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1.
To consume a beverage of some sort.
Technically it should only be used in reference to drinking beer because of the history of the phrase:
Back in the middle ages when beer was sold in ceramic cups they would be fitted with a small whistle which could be blown upon finishing ones beverage in order to alert the bar staff that you require another beer. Therefore to get a beer would literally be "wetting ones whistle"
Martin - oh look, I finished my lovely pint of ale. Best wet my whistle with another Broadside or London Pride.
Alexi - Get me some chillie nuts or prawn cocktail crisps while your there, you slag.
by MBAP88 September 19, 2008
 
2.
To consume a beverage of some sort to cleanse one's pallet.
water is a common choice or crunk juice.

Pronounced {whet-mi-wiss-el}
Brad: Oh boy, am I thirsty! I need something to Wet my whistle!

Carson: Want some water?

Brad: I sure do!
by Cselby January 26, 2007
 
3.
Basically, it means to do something you enjoy. Technically, it means you do something to moisturize your mouth, like drink something.
I'm going to drink a beer to wet my whistle.
I like that car, it really wets my whistle.
by DP2 September 01, 2006