The word nightingale has been around for a while and is in no way related to gale. In Middle English, it was spelled nightingal, nyghtyngale, nyghtgale, nightegale, nittingale, and several other variations using the yogh as a stand-in for the voiceless velar fricative sound. The central n slowly crept in over time, but wasn't originally there. This is all from Old English nihtegale, which literally meant "night-singer" (a reference to the noises the males of the species make, which in fact occur during both night and day). The nihte part is where night comes from: it traces to the Proto-Germanic root nahts, which is from Proto-Indo-European nokts. Gale, meanwhile, was a conjugation of the Old English word galon, meaning "to sing". That is thought to be from Proto-Indo-European ghel, "to yell".
Adam Aleksic is a sophomore studying linguistics and government at Harvard University. He also has disturbing interests in politics, vexillology, geography, board games, conlanging, and law.