For this sentence...
...your first choice in this example might be "We're trying to come up with our band's name." But that isn't correct. Your band doesn't have a name yet, so you can't say "our band's name". You have to say "a band name" or "a name for our band".
This is a similar problem to sentences like "I'm going out to buy my swimsuit this afternoon." If you haven't bought the swimsuit yet, you can't call it "my swimsuit". You have to say:
I'm going out to buy a swimsuit.