This isn't just Fallon being Fallon, however: Apparently it's a real, longstanding Late Night tradition. David Letterman's staff gave the pickle to Conan O'Brien when he inherited the show, and O'Brien gave it to Fallon when he took over.
The tradition's weirdness doesn't preclude sentimentality, though. O'Brien's staff reportedly included a note to Fallon's when they handed off the pickle:
"Whenever you're done – which won't be for a long time – make sure you pass it on to the next sap," the note read. "Knock 'em dead."