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, a rising sophomore studying linguistics and government at Harvard University, has been described as the internet's sixth most famous etymologist. He also has disturbing interests in politics, vexillology, geography, board games, conlanging, and law.