The Unseen Hand would lend itself to a handshake or a certain pattern of fingers drummed on ones back/leg/wherever.
The Nighthawks could use Caprimulgus [Wikipedia entry: A nighthawk is a nocturnal bird of the subfamily Chordeilinae, within the nightjar family, Caprimulgidae. Nightjars are sometimes referred to as goatsuckers from the mistaken belief that they suck milk from goats (the Latin for goatsucker is Caprimulgus).]
But then, both groups speak Thieves Cant (which in my head is a slightly more aggressive form of Cockney rhyming slang), so the Nighthawks' secret word would presumably some phrase in Thieves Can't which, being understood by their rivals, would need to change regularly.
Maybe this is why the Unseen Hand developed a tactile "password", which in turn influenced their choice of name?