Blinking was the regular closing and opening of one's eyes. To have something happen "in the blink of an eye" was for it to occur quickly. (DS9: "In the Cards" et al.) To not blink at taking an action was to do it without any hesitation or reservations. (DS9: "When It Rains...")
In 2267, Dr. Leonard McCoy noted that Hikaru Sulu and Montgomery Scott were likely drugged, as they were hardly blinking. (TOS: "Catspaw")
In 2366, Data taught his android "daughter" Lal how to blink. (TNG: "The Offspring")
Data's blink rate was regulated by the Fourier series. In 2370, he noticed that his "mother"'s blinks had the same pattern, which clued him in to the fact that she was really an android that had replaced the Human woman. (TNG: "Inheritance")
Not blinking could refer metaphorically to a refusal to be the first to give in. In 2375, Admiral William Ross told Kira Nerys she could blink after Senator Kimara Cretak gave in to the demand to remove plasma torpedoes from Derna. (DS9: "Shadows and Symbols")
In 2376, while Neelix played a shell game with tera nuts in an attempt to confuse The Doctor, Tom Paris warned the hologram not to blink. The Doctor assured Paris that his optical subroutines were up to the task. (VOY: "Live Fast and Prosper")
In 3189, during her debriefing by Starfleet personnel, Emperor Philippa Georgiou deactivated her holographic interrogators by rapidly blinking her eyes. After Kovich said that she had broken his holos, she replied, "Blinking at their harmonic rate disrupts the holo protocols, creating a reference loop that shuts them down." (DIS: "Die Trying")