Fig.: to have a drink, especially of liquor

After a dusty ride, he stopped in the saloon to wet his whistle

Dopo una cavalcata nella pianura arida e polverosa, il cow-boy si è fermato nel saloon della città per bagnarsi il becco

We've been walking a lot in such a hot weather. I need to stop, rest a while and wet my whistle

È da un bel po' che camminiamo e fa molto caldo! Ho bisogno di fermarmi un attimo e bagnarmi la gola

It looks like that "whistle" is an ancient slang for mouth or throat, which must be kept moist in order to speak, sing or, above all, whistle.
According to another theory - which has never been confirmed but is quite popular - pub patrons in old-time England, had a whistle embedded in their pub cup. When they needed a refill, they used the whistle to get service