1) Hightower Martial Syndicate / Godslayers: The campaign began with the PCs belonging to a mercenary guild (HMS) run by a dude with the last name Hightower. At the end of the first session, that NPC died and the PCs inherited his company. The guild was responsible for killing a "Living God" in that first session, a sort of sorcerer-king type. Regular NPC folks started calling them the Godslayers, and it was my hope the players would adopt it as their new guild name since the Hightower dude was dead. Unfortunately the campaign came to a premature end.
2) Voice of the Bold: Whereas the premise of the previous campaign was to thrust ownership of a merc guild onto the players, this group of players chose to start one of their own. VotB was literally the first thing a player suggested, and no one objected, so the players stuck with it. I always thought it was lame as hell, but w/e.
3) The Sentinels / Nightdarts : I approached session 0 a little differently with this group, and asked that they conceive of a group identity before coming up with individual character concepts. They decided to be a church operated special forces group. A few ideas got tossed around, they ended up going with the Sentinels. I decided there would be 4 other squads in addition to the PCs. To give this name a little more depth, I suggested the "sentinel" is a type of bird of prey in this world, and that each squad would be named after a subspecies of Sentinel. The subspecies names I suggested were Grasslancers, Skyfishers, Barnhawks, Redcrests, and Nightdarts. The PCs chose the last one.
when I DM'ed Princes of the Apocalypse for my friends they, at one point, were facing an enraged Umber Hulk....one of the party members dropped and played dead and they found that if they all did that the Umber Hulk ignored them and charged at another enemy that the group was facing....thus was born the Fainting Goats adventuring party
Usually the group I play with/run for doesn't bother with a group name... unless there are in-game reasons for one. The only exception I can recall is the all Bards game I ran years ago. Someone dubbed the party "Music and Mayhem"