Not a bad idea, and all very good points. Intelligently, the team name should rest on the shoulders of one character -- your mastermind -- who should pick from a group of competent, efficient, and above all subtle individuals to form 'the team' as necessary for each mission. If team names don't do it for you, though, think about it more as 'get this guy, who'll get a crew he can hang his rep on'.
Never thought about registering as a merc crew, though.