What sort of name should the character have?
You're in for a treat, my friend. The melee/unarmed gladiator is easily the most powerful build in the game, to my experience. New Vegas introduces a few new perks that totally step up your potential.
1. At character creation, choose "Heavy Handed" - a trait that will boost melee/unarmed damage at the expense of critical hit damage.
2. Choosing male or female doesn't matter, but the Legion takes you more seriously as a male, FYI.
3. Plan ahead for the perks you want. You're going to want to put points into both melee
and unarmed, regardless of which one you plan on using, as both skills have perks available that affect both attack types.
4. Perks to consider:
Toughness - You'll need a high DT, to survive the time it takes to close in on a ranged enemy.
Stonewall - You can no longer be knocked down. You receive +5 DT when you're wielding melee/unarmed weapons.
Super Slam - The centerpiece of this build. Attacks knock enemies to the ground. An enemy that is knocked down is an enemy that's not attacking you.
Terrifying Presence - Requires high Speech, but it's fun to use. It frightens opponents into cowering if you initiate combat from dialogue (i.e. after meeting Benny for the second time)
Piercing Strike - Another essential. Ignores 15 points of enemy DT. You'll be cutting through Deathclaws like they used to cut through you.
Unstoppable Force - Totally cuts through an enemy's block, if they're using melee. All attacks will land fully, and effectively.
Purifier - Does 50% more damage against "abominations" like Deathclaws, feral ghouls, and super mutants, but only with melee/unarmed. Awesome.
Paralyzing Palm - As mighty as it ever was...though, you won't need it as much with Super Slam in your repetoire.
Slayer - Increases melee/unarmed attack rate by 30%.
Any other perks are entirely up to you - experiment! I chose Confirmed Bachelor and Cannibal to fully flesh out my character. (you'll never run out of food again!) If you pick these perks, you're more or less unstoppable. Don't forget to get the DT implant from the New Vegas clinic. All you need to worry about is surviving long enough to get up close to your foe. Once you are, one or two hits is usually all it takes.
As for names, pick something simple, and ominous. I named mine "Ronin", because he considers himself a wasteland samurai.