“We're trying to come up with a name for our band.”
You are starting a band with a group of friends who are musicians. You're all sitting together and brainstorming ideas for band names. Your girlfriend sees you all sitting around and looking thoughtful, and asks what you're doing. This is your answer.
We're trying to come up with a name for our band.
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come up with (something)
To "come up with" an idea means to create it. You usually use "create" for physical things that you make, but you use "come up with ___" to describe creating an idea, a plan, a name, etc. Another example:
Where'd you come up with the idea to start a blog about English phrases?
a name for (something)
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.