I think it's good to come up with a number. Research a bit and see what the market price for what you offer or what people in a similar position get paid. If you think you're better than average, maybe increase the number slightly, if you think you're worse, leave it at average. "Naming a price" shouldn't hurt your chances, but does make an impression you're assertive and confident in regards to what you offer. So in that regard, it could actually increase them, as no one really wants or at least values a push-around or someone who isn't sure of his abilities.
I know it's hard when looking for that first job, and you want to do everything you can just to get it, but the concern you voiced is on point, I'd say. You don't want to just low-ball and hope you're going to get in because of that. The chances of that are namely quite slim and even then, the only employer who is going to make such a bad choice is a bad one, not in terms of personality, but in terms of his competence. Let's face it, making business decisions based on what other people tell you the price should be isn't a sign of a successful business operator, is it?
Plus, you say you don't care about money, but the fact is that if you do get in and see everyone else making twice the amount for the same job, you're going to feel bad, really bad. You're going to hate yourself for not giving out a realistic price, because, well, it's going to be hard for you to later negotiate and demand a higher wage. And that could then also have a negative influence on your work and how you feel in that working environment. You need to be assertive, but realistic. Don't name a ridiculous amount, but also don't shoot yourself in the foot by underestimating your worth. That kind of thing, at least in my experience, is also bad in principle, because people start competing for places with lower and lower wages, but working more and more. You just work yourself to death that way.