The airport shuts down by midnight almost every night. When the night winds down you take people home from the bars and restaurants, then the bar and restaurant workers go home. Then you get some stragglers. But I'd say there's business around until about 3am.
Last summer, all summer I made good money driving Uber Black doing Logan runs beginning about 4:15AM. I could get 3-4 jobs in by 9:00AM. About $90 each. I'd go out to Rt. 128 and back again. You can't do that with UberX because the rates are way too low. With Fasten I can because of the low commission but they don't have the work out there.