Thirsty Thursday is the Mother of all promotional nights. In fact, it is certainly not unique to baseball or sports marketing. Bars, restaurants, even nursing homes have Thirsty Thursday. But, Minor League baseball marketers seems to have turned it into a thing in and of itself. It dominates the landscape of Minor League Promotions like Devil's Tower.
The only seeming limitation to Thirsty Thursday are local ordinances. For example, in my original home state of Rhode Island, there are very restrictive laws about drink prices and "Happy Hour" specials. Outside of this virtually all team offers a drink and/or food special on Thursday.
Why Thursday? Probably because it rhymes. But, also because it is leading into the weekend's usually larger promotions. And finally, it is pretty easy, especially since you can probably get distributors to offer discount prices on the beers/liquors themselves.
Though, the lesson of Disco Night at Old Comisky in 1979 should always be remembered when mixing other promos with happy hour.