Perhaps, but I live here and I'm doubtful. Not that I'm one...

Have you called Avis? Your options are somewhat limited in that Logan is the only Avis 24 hr location hereabouts. The other stations at Government Center (apart from Logan closest to the airport but on the other end of the Sumner Tunnel near Quincy Market, North Station/Boston Garden, Downtown), the Back Bay (near Copley Place, Newbury St, etc.) and Harvard Square (in the Charles Hotel in Cambridge) are daytime only. The Back Bay location opens at 7 on Saturday mornings, the other two at 8 am.

Meaning you either pay the extra (assuming it costs more) to return the car at Logan at 4 am - and it can be done: I did so last week for a 5.45 am flight - or return the car tonight and pay for the T back to your hotel if it'll get you there (or a cab if it won't) and a cab back to Logan in the morning because nowhere does the T start running until about 5:45 am, meaning getting to Logan by 6:15 am soonest.

Not knowing when a "very early flight" is (9 am for me, I wish) nor where you're staying tonight it's impossible to give you an exact response for your situation, but it sounds like you pay either way. A cab to Logan from most anywhere downtown is going to be at least $20 at that hour, from Cambridge or the 'burbs more. Only Avis can say what they'd charge so it sounds like you've got to ask them.