External is best, but what you could do is use CrashPlan because it allows you to backup to any other computer on your home network (or even a remote network) for free, then have a time machine or carbon copy cloner drive setup on your mini.
This is more or less how I handle my setup.
Can I use mini as a Time Machine backup for the MBA?
No, you're not doing what the OP asked. You're backing up to an external drive, which is what was suggested in the first two responses. You're not backing up the MBA onto the Mac mini, but rather onto an external drive.Yes. I do this.Can I use mini as a Time Machine backup for the MBA?
My mini has a number of external drives connected, one of which houses TM backups for my 3 machines.
No, you're not doing what the OP asked. You're backing up to an external drive, which is what was suggested in the first two responses. You're not backing up the MBA onto the Mac mini, but rather onto an external drive.
No. You can't have the Mac mini and another computer both trying to control the mini's internal drive. You have to use an external drive for backups.So partition the mac mini drive and he should be able to TM on that partition ?