Extremely. Basically critical.
If I don't feel like my character is "needed" I immediately lose interest in them. I don't think this is very normal in that I see lots of people wanting to make the same generic good-looking vampire/noblelady etc. as everyone else on most games, but that's how I feel about it.
It's not just about being "needed" by other people; I want to feel like I'm filling in a niche role and helping to balance the demographic, sure, but I also want to feel ... special? Original. While I realise there's no new idea under the sun, when I come up with a character concept I like to think it's the greatest idea anyone's ever come up with and when I then roll in to discover there are ten others exactly like me, I get bummed out.
Private confession time: on the 100 MUSH, I rolled an Earth Skills nerd and when I realised @GirlCalledBlu had actually beaten me to it I almost wanted to reroll. This is of course not her fault at all, just a quirk of mine. I have a need to be the only one doing a certain thing, but I was able to carve out my own niche in other ways, despite that.
This is part of the reason why I usually like playing underdogs, 90lb weaklings, less conventionally attractive characters, older/mature characters, poor people, etc.
I didn't play Arx for very long (not because I don't think it's a very good game, it just wasn't my theme of choice) but when I did I specifically went for a commoner. I like being a foil for others and don't really see the appeal of playing a lord in a game where you can't actually lord over anyone, since everyone else is a lord/lady too.
When rolling into a new game I tend to ask staff/players what type of character they think is currently "needed". Although I think it's very kind of them to do so, I always get a little bit annoyed when someone tells me "just play what you like to play!" Because what I like to play is, in fact, whatever happens to be most needed at any given point.