There are five things that interest me in playing a character: effectiveness, theme, costume, power appearance/coolness, name.
If the character is just bad at defeating foes or doing what I want, not going to have a fun time. Doesn't have to be the most min-maxed or anything, just... not suck.
This can be tricky with Archetypes, though Become devices help somewhat.
There needs to be some cool concept that brings the character together, though this can cause headaches when the concept is hard to do in the game. For example, I had a water-themed character that worked well with some of the newer abilities in freeform (wind, earth quicksand ability, as well as metallic blue ooze travel).
It's harder (and, IMO, less fun) to do creative themes in Archetypes, but oh well.
Some themes just fail for a theme purist -- I tried to do Doc Wormwood, a CoX transplant with a 'filled with radiation' theme. Where things broke down is that, in strict CO game terms, 'radiation' = particle damage, possibly with toxic/fire side effects.
But the only particle damage is in Power Armor, and eyeblasts/chest beams didn't really fit the concept. I tried green fire and toxic attacks, which look awesome, but... I kept being bugged by the lack of mesh with the game 'truths' (I was also annoyed that electric passive was the only passive that protects against particle damage).
So I remade him as some gonzo 'uber aoe' guy (epidemic + flashfire + sigils of primal form + other stuff I forget), which made him hugely effective. And then lost interest.
This one is a no-brainer. If my character doesn't look really neat to me, I lose interest.
Related to costume and theme, powers that have cool effects or appearance intrigue me. Things like ebon rift, summon shadows, and orbital cannon? WOO!
A lot of the weapon abilities don't thrill me because I find the weapon models clunky and the damage application ... dull. Blam blam blam. Meh.
And, finally... Name
The topic of the thread!
What ties all of the above together and makes me keen on playing a character is a name. Some name that just makes me go 'HA!'