Short answer:
It seems quite possible for hypothetical habitable exomoons to have days as long as two Earth weeks. Day lengths of several Earth months or years seem to be less plausible.
Long answer:
Alexander's answer is pretty good as far as it goes.
But according to my rough calculations, a planet orbiting at 2.4 AU from the Sun would have a year about 3.7180 Earth years, or 1,358.0228 Earth days, long, and its hypothetical moon could have a month/day no longer than about 150.8914 Earth days long, not the 277 days that Alexander calculates. There is another complicating factor which Alexander did not allow for in his calculations.
There have been a lot of other questions about habitable moons of gas giant planets in the habitable zones of stars, and it is a good idea to refer to those questions and answers to see if they have any useful information, as I state in my answer to this question:
How long will it take to discover they live on a moon and not on a planet? 1
And I gave links to two earlier questions about habitable exomoons.
The article "Exomoon Habitability Constrained by Illumination and Tidal heating" by Rene Heller and Roy Barnes Astrobiology, January 2013, discusses factors affecting the habitability of exomoons.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3549631/2
And it suggests that the longest possible day for a hypothetical habitable exomoon would be less than, for example, a single Earth year long.
It is assumed that the vast majority of habitable exomoons would be tidally locked to their primaries, rotating at the same rate as they orbited those planets, and thus keeping one side facing the planet at all times and other side facing away at all times. Thus the moon's month, or orbital period around the planet, should be same length as it's day, the time the moon takes to rotate through 360 degrees.
Thus I tend to call it the month/day of the moon, since as the moon orbits and the planet and also rotates it will rotate in relation to the star or sun in the solar system and thus the sun will rise and set and a spot on the surface of the moon will experience a period of daylight and a period of night during the moon's orbital period around the planet.
On moons, however, tides from the star are mostly negligible compared to the tidal drag from the planet. Thus, in most cases exomoons will be tidally locked to their host planet rather than to the star (Dole, 1964; Gonzalez, 2005; Henning et al., 2009; Kaltenegger, 2010; Kipping et al., 2010) so that (i.) a satellite's rotation period will equal its orbital period about the planet, (ii.) a moon will orbit the planet in its equatorial plane (due to the Kozai mechanism and tidal evolution, Porter and Grundy, 2011), and (iii.) a moon's rotation axis will be perpendicular to its orbit about the planet. A combination of (ii.) and (iii.) will cause the satellite to have the same obliquity with respect to the circumstellar orbit as the planet.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3549631/2
The longer the month/day is, the hotter the day side will get, and the colder the night side will get. The longest possible day or night for a planet would be if that planet was tidally locked to its sun, and thus had eternal day on the near side and eternal night on the far side of the planet.
There is a fear that a planet tidally locked to its sun would lose its atmosphere and water because the hot air and water vapor from the day side would flow to the night side and condense and freeze until everything was frozen on the night side.
If that was the case, a planet that had a sufficiently long day period would have almost all of its water and atmosphere frozen on the night side during the long night. Only the water and atmosphere that was melted and sublimated at dawn would exist as a thin atmosphere that sublimated at the same rate as it froze out.
On the other hand, it is possible that the circulation of air and water between the light and the dark sides will transfer enough heat to the dark side to keep the air and water unfrozen.
This pessimism has been tempered by research. Studies by Robert Haberle and Manoj Joshi of NASA's Ames Research Center in California have shown that a planet's atmosphere (assuming it included greenhouse gases CO2 and H2O) need only be 100 mbs, or 10% of Earth's atmosphere, for the star's heat to be effectively carried to the night side.[74] This is well within the levels required for photosynthesis, though water would still remain frozen on the dark side in some of their models. Martin Heath of Greenwich Community College, has shown that seawater, too, could be effectively circulated without freezing solid if the ocean basins were deep enough to allow free flow beneath the night side's ice cap. Further research—including a consideration of the amount of photosynthetically active radiation—suggested that tidally locked planets in red dwarf systems might at least be habitable for higher plants.[75]
https://en.wikipedia.org/wiki/Planetary_habitability#Other_factors_limiting_habitability3
So at the present time it seems possible that even a tidally locked planet could be habitable, and thus there doesn't seem to be any known limit based on freezing out the atmosphere to how long the day and night of a habitable exomoon could last, which is good for your desire to have it as long as possible.
According to "Exomoon Habitability Constrained by Illumination and Tidal heating"
The synchronized rotation periods of putative Earth-mass exomoons around giant planets could be in the same range as the orbital periods of the Galilean moons around Jupiter (1.7–16.7 d) and as Titan's orbital period around Saturn (≈16 d) (NASA/JPL planetary satellite ephemerides)4. The longest possible length of a satellite's day compatible with Hill stability has been shown to be about Pp/9, Pp being the planet's orbital period about the star (Kipping, 2009a)
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3549631/2
So they estimate that a habitable exomoon might have a month/day as long as maybe 17.0 Earth days. But what is really important is:
The longest possible length of a satellite's day compatible with Hill stability has been shown to be about Pp/9, Pp being the planet's orbital period about the star (Kipping, 2009a)
The source, Kipping 2009a, seems to be:
Kipping D.M. Transit timing effects due to an exomoon. Mon Not R Astron Soc. 2009a;392:181–189.
https://arxiv.org/abs/0810.22434
According to Alexander's answer, if the exomoon was orbiting a planet orbiting a star as massive as Sol (the sun) at a distance of 2.4 AU, assumed to be the other limit of the habitable zone, the habitable moon could have a month/day of 277 Earth days or 0.76 Earth years.
If the year of the planet has to be at least nine month/days of the moon long in order for the moon to have a stable orbit, the year of the planet would have to be at least 2,493 Earth days, or 6.825462 Earth years.
There are many different scientific estimates of the habitable zone of the Sun, or of a star that is exactly like the Sun. Some estimates give the some a very narrow habitable zone and other estimates give it a very broad habitable zone.
Since you are interested in the longest possible month/day of your moon, and thus the longest possible year for the planet orbiting it's star, lets calculate it for various outer edges of the Sun's habitable zone.
If the hypothetical moon's planet orbits a star exactly like the Sun at a distance of exactly one AU, the planet will have a year exactly one Earth year long, and the longest possible length of a month/day of a habitable moon of that planet would be one ninth of an Earth year, or about 40.5833 Earth days.
According to this paper:
Hart, M. H. (1979). "Habitable zones about main sequence stars". Icarus. 37: 351–357. Bibcode:1979Icar...37..351H. doi:10.1016/0019-1035(79)90141-6.
The outer edge of the Sun's habitable zone is only 1.01 AU from the Sun. According to my rough calculations, a planet orbiting at that distance would have a year about 1.01503 Earth years, or 370.7424 Earth days long, and its hypothetical moon could have a month/day no longer than about 41.1196 Earth days long.
According to this article:
Vladilo, Giovanni; Murante, Giuseppe; Silva, Laura; Provenzale, Antonello; Ferri, Gaia; Ragazzini, Gregorio (March 2013). "The habitable zone of Earth-like planets with different levels of atmospheric pressure". The Astrophysical Journal. 767 (1): 65–?. arXiv:1302.4566. Bibcode:2013ApJ...767...65V. doi:10.1088/0004-637X/767/1/65.
The outer edge of the Sun's habitable zone is only 1.18 AU from the Sun. According to my rough calculations, a planet orbiting at that distance would have a year about 1.2818 Earth years, or 468.1803 Earth days long, and its hypothetical moon could have a month/day no longer than about 52.0200 Earth days long.
According to this article:
Kasting, James F.; Whitmire, Daniel P.; Reynolds, Ray T. (January 1993). "Habitable Zones around Main Sequence Stars". Icarus. 101 (1): 108–118.
The outer edge of the Sun's habitable zone is 1.37 AU from the Sun. According to my rough calculations, a planet orbiting at that distance would have a year about 1.6035 Earth years, or 585.6943 Earth days long, and its hypothetical moon could have a month/day no longer than about 65.0771 Earth days long.
According to this article:
Kopparapu, Ravi Kumar (2013). "A revised estimate of the occurrence rate of terrestrial planets in the habitable zones around kepler m-dwarfs". The Astrophysical Journal Letters. 767 (1): L8. arXiv:1303.2649. Bibcode:2013ApJ...767L...8K. doi:10.1088/2041-8205/767/1/L8.
The outer edge of the Sun's habitable zone is 1.68 AU from the Sun. According to my rough calculations, a planet orbiting at that distance would have a year about 2.1775 Earth years, or 795.3423 Earth days, long, and its hypothetical moon could have a month/day no longer than about 88.3713 Earth days long.
According to this article:
Spiegel, D. S.; Raymond, S. N.; Dressing, C. D.; Scharf, C. A.; Mitchell, J. L. (2010). "Generalized Milankovitch Cycles and Long-Term Climatic Habitability". The Astrophysical Journal. 721 (2): 1308–1318. arXiv:1002.4877. Bibcode:2010ApJ...721.1308S. doi:10.1088/0004-637X/721/2/1308.
http://iopscience.iop.org/article/10.1088/0004-637X/721/2/1308/meta5
The outer edge of the Sun's habitable zone is 2.00 AU from the Sun. According to my rough calculations, a planet orbiting at that distance would have a year about 2.8284 Earth years, or 1,033.0829 Earth days, long, and its hypothetical moon could have a month/day no longer than about 114.7869 Earth days long.
According to this article:
Ramirez, Ramses; Kaltenegger, Lisa (2017). "A Volcanic Hydrogen Habitable Zone". The Astrophysical Journal Letters. 837: L4. arXiv:1702.08618 [astro-ph.EP]. Bibcode:2017ApJ...837L...4R. doi:10.3847/2041-
http://adsabs.harvard.edu/abs/2017ApJ...837L...4R6
The outer edge of the Sun's habitable zone is 2.4 AU from the Sun. According to my rough calculations, a planet orbiting at that distance would have a year about 3.7180 Earth years, or 1,358.0228 Earth days, long, and its hypothetical moon could have a month/day no longer than about 150.8914 Earth days long. That is the same distance from the Sun that Alexander used to calculate a month/day of 277 Earth days.
However, this seems to involve atmospheric hydrogen concentrations of 1 % to 50 %, which do not seem compatible with an oxygen rich atmospheres suitable for humans.
According to this article:
Fogg, M. J. (1992). "An Estimate of the Prevalence of Biocompatible and Habitable Planets". Journal of the British Interplanetary Society. 45 (1): 3–12. Bibcode:1992JBIS...45....3F. PMID 11539465.
The outer edge of the Sun's habitable zone is 3.00 AU from the Sun. According to my rough calculations, a planet orbiting at that distance would have a year about 5.1961 Earth years, or 1,897.8946 Earth days, long, and its hypothetical moon could have a month/day no longer than about 210.8771 Earth days long.
According to this article:
Pierrehumbert, Raymond; Gaidos, Eric (2011). "Hydrogen Greenhouse Planets Beyond the Habitable Zone". The Astrophysical Journal Letters. 734: L13. arXiv:1105.0021 [astro-ph.EP]. Bibcode:2011ApJ...734L..13P. doi:10.1088/2041-8205/734/1/L13. Cite uses deprecated parameter |class= (help)
http://adsabs.harvard.edu/abs/2011ApJ...734L..13P7
The outer edge of the Sun's habitable zone is 10 AU from the Sun. According to my rough calculations, a planet orbiting at that distance would have a year about 31.6227 Earth years, or 11,550.218 Earth days, long, and its hypothetical moon could have a month/day no longer than about 1,283.3575 Earth days long.
But this last calculation involves planets with significant amounts of hydrogen in their atmospheres, equal to or greater than Earth's total atmospheric pressure, which would not be consistent with a breathable oxygen rich atmosphere for humans.
https://en.wikipedia.org/wiki/Circumstellar_habitable_zone8
Another way to change the possible length of the year of the planet and thus of the month/day of the moon, is to change the mass and thus the luminosity of the star in the system.
A relatively small change in the mass of the star can produce a much larger change in the luminosity, and thus in the distance of the habitable zone, and thus in the length of the years of planets in the habitable zone, and thus in the maximum possible length of the month/days of moons orbiting those planets.
And a relatively small change in the mass of the star can produce a much larger change in the rate at which it uses up is nuclear fuel and thus the time it depends on the main sequence stage of its life before becoming a red giant star and then a white dwarf star.
And if you want your hypothetical moon to have multi celled lifeforms, or an oxygen rich atmosphere breathable for humans, or intelligent natives, or most of the other things which are usually needed to make a world interesting in science fiction, you will want it to be billions of years old and thus you will need the moon's star to be of a spectral type capable of remaining on the main sequence for several billion years.