Why does a 22 year-old applying for a student credit card with Discover need a co-signer?

No, there's really no way to convince them. Although you've been responsible with your personal finances, there's nothing you listed (other than the student loans) that would create any credit history. You also don't mention what your current income is (future potential income doesn't count). So from the lender's point of view, there's a fair amount of risk.