So apparently, there's been a longstanding struggle of my Pathfinder game's GM.

He wants the PC party to give the adventuring group a name. Which the rest of the players have never, ever done. When another (NPC) party approached and spoke with us in the last session, they introduced themselves as The Backdoor Bandits. The PC speaking to them said that we were an unknown group.

So now, I kind of want to help name the group. But it's difficult, because we basically just met each other at the beginning of this campaign (with a few exceptions: myself and another player were the same characters from the last campaign carried over, and two half-orc characters were written to have been working together for a while now). Two characters (myself and the Mystic Theurge) worship Desna, two (the half-orcs) worship a god of war and death, one's a kobold who worships dragons, and one is a ranger that has no deity. So there's no commonality there. The missions within the campaign so far have been investigative ones, but it's leading us into entering The Tomb of Horrors.

I was thinking of something like The Shadow Piercers, something that indicates that the party goes to dark and scary places and stabs and arrows to death whatever threatens them in it.

... Name an adventuring group? ... That's just freakin' bizarre. All I have to say on that is good luck finding something for everybody to agree on, you'll need it.

The first question I suggest is, what does the group need a name for? Tax purposes? Advertising? Reservations at the next inn? Future employees? Is the name to give the group a sense of identity for themselves, or to fix its place in the social and legal structures of the world you're in?
You look through old chronicles and sagas, and groups get called things like, "Robin Hood and the outlaws," or Roland and his army." It's only in literature that you get "R.H and his Merry Men" or "Roland and the Twelve Paladins."
By the time of the Renaissance you got mercenary bands with slightly more creative names (not memorable enough to come to my mind just now), but mostly named after whoever was in charge of them, or the colour they wore, or what they did.
Shadow Piercers Security Force? (Commonly called Shadow Piercers?)

I believe it's so the group can have some level of infamy.

As for getting buy-in on the name, I was figuring my character would just introduce the group as that name the next time we meet somebody new, and then the name might stick.

It would amuse me greatly if it resulted in everyone calling the group a differing name, so we collectively have 6 aliases.

The Cleanup Crew
The Derelicts

A name like that is usually earned, I'd think, not chosen. Unless it is some official organization formed and gathered for some purpose, then the purpose lends itself to the name.

How about... The Adventurers

In genre example group names: The Fellowship of the Ring, the Night's Watch, the Band of the Red Hand, the Brotherhood of the Wolf, the Order of the Stick

Or they could reclaim an insult like the Rejected - after all the Yakuza are named after a losing hand of cards

This. If you're not named by your purpose (see: Fellowship of the Ring), your name is given to you by outsiders--it's what they've come to call you. In most cases, this will be based on some great deed you accomplish--so if you come out the backside of the Tomb intact, then you discover that people call you the Delvers of the Tomb, and that's what the bards refer to you as in their retelling.

I'd go along the lines of telling the DM, "Well, when we hear people talk about us, what do they call us?"

In real life, you very rarely are allowed to give yourself a nickname. It's usually a moniker that's bestowed upon you. It seems like the DM really wants a name so he can start to give fame/infamy etc. If the rest of the group isn't as into doing that, I'd go the route of having an NPC (read: DM) name the group and just move on.

I do have one campaign where I am into the fame game. I'm a masked archer who is always paying bards to spread the name of The Black Rose. I leave trademark tokens behind for major kills, never miss an opportunity to do something that would help my fame. The rest of the group has zero interest in that, which is fine because when our group does something cool, when I tell the tale I minimize their contributions and maximize mine. It can be fun RP.