Wet Your Appetite
To whet your appetite, with a "h," is the correct spelling of this phrase. It's problematic because the equivalent expression, wet your whistle, which meaning to have a drink, is accurate when written without a "h." It makes sense that people would believe that wet your appetite is also accurate given that you utilize one for food and the other for liquids.
Actually, no, since the two terms do not have the same meaning. Whetting your appetite, on the other hand, indicates virtually the exact opposite of wetting your whistle—to stimulate it or, when used metaphorically, to heighten your excitement.
It makes far more sense in the context of the idiom since to whet something is to sharpen it. This is another word that has been around for centuries but has lost favor in recent years because few people today visit the blacksmith to sharpen their swords. That, together with the widespread use of the comparable idiom wet your whistle, accounts for the shift away from whet to its more popular homophone.