In the two groups in which I'm a player, I went out of the way to make sure we had a group name, and it's added a lot to our roleplaying.
One group is called The Scaled Company, after the very unlikely defeat of a Tyrannosaurus Rex when the group was only 3rd- and 4th-level. We skinned the beast and made several pairs of masterwork boots, which we give to new and replacement members as a sign they've earned our trust.
Another group is The Blood Griffons. My half-ogre named us when we needed a team name for the Champion's Belt event in Greyhawk. Technically, he spelled it "Blud Grifuns," but one of the other party members corrected it surreptitiously. The name is a variation on the half-ogre's father's company of mercenary ogres, The Black Griffons. (The father, BTW, is human, which is a longer story.)