You're paired with a sword so important that it needs a name.

LAAAAAZER SWOOOORD!

Mr. Stabby

In reality if I was a guy who'd need to be carrying a sword to kill I’d probably call it "Fucker" or something to that effect.

" Mr. Stabby "

Zan-Driver. 
It's a more correct version of "San-Driver" which was the name of a sword I thought up for an awful comic I wrote when I was 13. 
 
Other possibilities would be "Weird Death" or "Sword of Strings".

" I call it My Thrusting Manhood. Imagine hearing the words, "Prepare to feel the wrath of My Thrusting Manhood!" right before you die. There's a specific part of the afterlife where everybody just makes fun of you, and this gets you there. "

" Lucy. Inanimate objects should have female names. For instance, my computer is Jane, my old slow computer was Bessie. "

The Sword of a Thousand Truths.

dancing death

The Sword of Truthyness

Compensating .  

If my name were Mac, I would name it Cheese.

Terminus Est or...PoopDickPappy

My Willy?
 
That way, I can say that I can attack things with my willy and be really awesome.

FACECUTTER
 
SPLEENSPLICER
 
HURTBRINGER

Rock
cause its gonna be rockin yo shit up in down the block

the big sword.

Bitch 
 
Mildly sexist but catchy.

" Blade of Fart "

Swordlet.

Shirvalla!

" Cutty McStabenslash. "

" The Sword of a Thousand Truths. "

Bringer des Blutes

Killerham Lincoln.
 Also, it's a member of Lincoln Force

Rathantoras, although if i were a woman i would call it my broad sword... ha.

Keen Eddie

" Skittles. "

Birdsong.

Carlo Von Sexron.

"The talking blade", if it can't talk. If it can, then "The quiet blade".

The Seaward.

Vendetta.

" Zach? "

Black Lodge Bob.

The long stainless steel thing with a sharp edge that's extremely good for cutting off people's heads 

Oni-Ten

Common Sword. 
 
It would be storebought, and not at all important. I'm defying the premise of your thread!

+2 Sword of Hack and Stab.

Y-Calibur. Get it? 

Iron Cabbage

Mike

Battenwarg

I would deny it a name, rather leaving the people to decide. After years of conquest, my sword and I will become either famous or infamous depending on our actions, and through this shared history it will earn its name.

iSword

Training Stick
Wooden Sword
Iron Sword
 
You know, those crappy swords in RPG's that you first train with in your village before bandits or mysterious strangers come and burn it down and you vow revenge. Or something like that.

Shiny Weapon Of Rapid Decapitating
or something like that

Toothpick.

" Lucy. Inanimate objects should have female names. For instance, my computer is Jane, my old slow computer was Bessie. "

Vagina McDuphrey
 
No one takes a guy with a sword named Vagina seriously.  Until they're dead of course.  The McDuphrey gives it a police detective/Celtic vibe.

Keanu.