Is it possible for a planet to be heated and illuminated by a black hole due to Hawking radiation at the same intensity as by a star?
What mass and size should a black hole have to produce the same amount of radiation as a star? How long it can be in such a state? What will be the black hole's size and the size of the habitable zone around it? Can a planet orbit this black hole without being ripped apart by tidal forces?
Essentially, can a planet orbit stably in a black hole's habitable zone?
This scenario is quite problematic for two main reasons: evaporation and peak wavelength.
We can make a rough estimate of the properties of the Hawking radiation coming from the black hole. First, let's start with the luminosity. Since $L\propto M^{-2}$, where $L$ is luminosity and $M$ is the mass of the black hole, it turns out that $$L=9.01\times10^{-29}\left(\frac{M}{M_{\odot}}\right)^{-2}\text{ Watts}=2.34\times10^{-55}\left(\frac{M}{M_{\odot}}\right)^{-2}L_{\odot}$$ where $M_{\odot}$ and $L_{\odot}$ are the mass and luminosity of the Sun. You need a very low-mass black hole to produce a significant amount of light. In fact, for a black hole to produce one solar luminosity worth of power, its mass must be about 960 kg. The big problem? Such a tiny black hole would evaporate in about 75 nanoseconds (and even during that time, the amount of optical light is' producing will be small - see below). You can prolong its lifetime by increasing its mass - the evaporation timescale is $\tau\propto M^3$ - but this will in turn decrease its luminosity, and so for the flux to be enough to make a planet habitable, you need to have you planet be closer to the black hole, which could be dangerous if the black hole is actively accreting matter.
The other major issue is that the peak wavelength of the radiation won't be in the visible band. A black hole's temperature is inversely proportional to its mass, and its peak wavelength $\lambda_p$ is inversely proportional to its temperature. We then have the relation $$\lambda_p=5.87\times10^{12}\left(\frac{M}{M_{\odot}}\right)\text{ nm}$$ and for our tiny, 960-kg black hole, the peak would be far, far, into the gamma ray portion of the spectrum - not great for life. For comparison, visible light has a wavelength of about 300-700 nm, and you'd need a black hole about 1% the mass of the Moon to produce optical Hawking radiation.
Others have talked about the possibility of energy from infalling matter in the black hole's accretion disk. Let's think about this a bit. There's a relationship between the maximum allowed luminosity - the Eddington limit - and the black hole's mass: $$L_{\text{Edd}}=1.26\times10^{31}\left(\frac{M}{M_{\odot}}\right)\text{ Watts}^{-1}=3.37\times10^4\left(\frac{M}{M_{\odot}}\right)L_{\odot}$$ Is this significant? Well, yes, definitely. But there are problems:
From Hawking radiation? No.
The Hawking radiation emitted is inversely proportional to the black hole's size. To make the black hole glow with enough light to be as bright as a star from Hawking radiation alone, it would need to be very small.
The problem with very small black holes is they also have very short lifetimes due to the Hawking radiation robbing them of energy, and thus mass ($E=mc^2$ after all). These small black holes eventually have a runaway of Hawking radiation and explode. This is the ultimate fate of all black holes in our universe, but on very large time scales.
As other answers have mentioned, a black hole that can emit photons in the optical EM range will have a mass of a small fraction of the Moon (~1%), and live for a very long time (~$10^{40}$ years). However, in order to emit such relatively high energy photons for such a long time, a black hole this size must burn VERY, VERY slowly. The SI unit for delivery of energy is the watt, and is roughly the same watt as a common household lightbulb.
1% of lunar mass is around 7e20 kg, and this would have an emitted power of ~7.3e-10 watts. Such a black hole would still be far too weak to warm any planets in orbit. The received flux of the planet Earth is, for reference, is around 1000 watts/m2, and this is an infinitesimal fraction of the Sun's total power output of 3.8e26 watts. This measure is called irradiance, and essentially tells you how many photons are passing through any given area at any given time.
If we assume the black hole is emitting photons at 700nm (a reddish color that is good for photosynthesis) at this wattage, it will emit 2.3 billion photons per second (per the Planck-Einstein relation and the definition of the watt as a joule-per-second). This may sound like a lot, but a 100 watt incandescent lightbulb is emitting $10^{32}$ photons per second, so your black hole is going to be extremely dim.
Irradiance is subject to the inverse-square law, so as you move away from the source, the received energy drops by the square of the distance. If your starting power is 1, and you double the distance, you now get 1/2 the power, triple the distance you get 1/9 the power, etc... Since your black hole is already only emitting less than a nanowatt, it only gets worse from there. Even if your black hole sits on the surface of your planet, it won't be able to warm the surrounding area, much less the entire planet.
However, it is possible for a planet in a black hole system to be illuminated, but only if there is a source of gas. Gas falling into a black hole can form an accretion disk, where the speed of the orbiting gas can release radiation through Bremsstrahlung and friction. This is why, for example, we were able to take the picture of M87*. What we were imaging was not the black hole itself, nor its Hawking radiation, but the light off of its accretion disk.
Unfortunately, the accretion disk needs to be replenished, so you will need a source of gas for it. In addition, the presence of this source of gas is likely to make orbits around your black hole unstable, which is not good news for your planet.
However, it is not impossible from a scientific standpoint, just less probable, and rather unlikely, that your planet will stay in a stable position long enough for life to evolve into anything interesting.
Keep in mind as well, that most black holes form as a result of supernovae, meaning that any planets around the star that explodes will be sterilized and probably vaporized by the creation of the black hole in the first place.
HDE 226868's answer covers the main points, but there is an additional way the environment around a black hole could heat a planet.
If the planet is close enough to the black hole that it experiences significant time dilation then the planet could get warmed by blueshifted cosmic background radiation. In order for this to work the black hole would need to have a high spin parameter, so that the innermost stable orbit is close enough to the black hole. You'd also need to have a supermassive black hole in order to have sufficiently weak tidal forces that the planet does not get ripped apart.
See the paper by Opatrný et al. (2016) "Life under a black sun" for more details. They calculate that the cosmic background radiation would heat Miller's planet in the film Interstellar to around 890°C, without taking the additional radiation from the accretion disc into account. As they note:
Thus, the tidal waves observed on the planet might be, e.g., of melted aluminum. Moreover, the astronauts would be grilled by extreme-UV radiation.
If you're not dead set on it being Hawking radiation, you might want to look into quasars. Basically, imagine a blackhole that is gobbling what's around it, and the whole thing radiating humongous amounts of energy in the process:
The brightest quasar in the sky is 3C 273 in the constellation of Virgo. It has an average apparent magnitude of 12.8 (bright enough to be seen through a medium-size amateur telescope), but it has an absolute magnitude of −26.7. From a distance of about 33 light-years, this object would shine in the sky about as brightly as our sun. This quasar's luminosity is, therefore, about 4 trillion (4 × 1012) times that of the Sun, or about 100 times that of the total light of giant galaxies like the Milky Way.
To put 33 light-years in context, Alpha Centauri is a bit over 4 light-years away from the Sun.
Anyway, with a bit of handwaving, it seems like you could imagine some earth-sized planet floating in deep space. Perhaps a galaxy crashed into another galaxy, resulting in the planet getting ejected out of its original star's vicinity. A quasar ignited when the two galaxies' cores crashed into one another, and the starless planet is now getting showered by visible light.
