What's your favorite drink? (This would typically be used for hard alcoholic drinks, not wine or beer)
What's your favorite thing to drink? (Open ended question. Milk, whiskey, wine, coffee, Red Bull, water from a spring on my grandfather's land, etc. i.e. most things are something you can drink, but not often called "drinks")
"Beverage" is a more official word and tends to make me think of a category. "Nestlé is the world's largest food and beverage company" My answer to "What's your favorite beverage?" would be "What do you mean by beverage?"
As far as Bramwell's usage of beverage, I would use "hot beverage" to mean tea, coffee, cocoa.