When I was younger, there was more of a tendency of naming our adventuring companies, but as I got older, it kinda faded into the background. We'd do it sometimes, but it wasn't really a priority.
The Champions of Baldwar Mountain - My first adventuring group, back when I was 7 years old. We defended a dwarven kingdom and its neighboring human ally from assaults by drow, duergar, orcs, goblins, evil wizards, undead, vengeful fey, evil cultists, and barbarian hordes. Very character-centric, with many sidetreks for personal reasons.
The House of Four - Retainers of a mysterious noble who delivered his orders through various intermediaries. We took on A-Team-style jobs while waiting for orders from the boss, with the campaign culminating in a strong arc featuring our master in person. Had a whole lot of politics and intrigue, along with some mass battle.
The Graveyard Shift - My friend ran a long undead-themed campaign back in high school, so we took this tongue-in-cheek name for ourselves. Very much beer and pretzels.
Scavenger Sons - The name my group took the last time I ran a mercenary themed campaign.