|"You should let me add a boon die to this|
attacl because my character has the
high ground and also the wind is
blowing in a favorable direction."
The phrase comes from Grima Wormtongue, the advisor in Lord of the Rings who pours rhetorical poison in Theoden's ear. (Although, to be honest, the fact that his name is Grima Wormtongue should have been a clear indication that he was not a guy you could trust with matters of state.)
In a gaming context, wormtonguing is when you're trying to convince the Dungeon Master that the course of action you're proposing is reasonable or even that your success is probable. You're wormtonguing when you're lobbying for some sort of bonus to your roll because of circumstances you see as favorable to whatever your character is attempting.
It seems like many Dungeon Masters hate being wormtongued; I've seen complaints about wormtonguing-esque behavior, although I've never seen it explicitly defined as such. The slang phrase Kevin and his group gave to this tactic hints at its slyness, selfish guile, and potential duplicity; that said, they clearly used the phrase in good humor. You could roll your eyes at a wormtonguing player, but it was accepted as a matter of course and as an expected part of playing the game as a group. You could laugh about a particularly bald-faced attempt at wormtonguing after the fact, but its use was enshrined as part of their social contract.
Personally, I love it. To me, when a player is attempting to sway things to their advantage, it's a sign that they're engaged with the game. They're thinking about the world, they're considering their character's goals and desires, and they're invested in the situation and outcome currently unfolding at the table. I suppose it could become wheedling, whining, or demanding when used by an unpleasant player, but you shouldn't be playing with that kind of person anyway. In my eyes, a good lashing from the wormtongue most often indicates excitement about the game. It's a sign that things are working as intended.
So, yeah, wormtongue me, baby.