Apparently I’ve never done this before. Time to rectify that.
You know how sometimes two words that are spelled the same but have nothing in common turn out to be from different sources? Yeah, well, this is not one of those times. Spring the season, which first showed up in the fifteenth century, is from the verb to spring, like something that would spring up. Before it was spring, it was just called Lent (lencten in Old English), and in the fourteenth century they started calling it “springing time”, when plants would “spring” from the ground. Although really, it doesn’t seem like they do much springing… Anyway, all versions of spring come from the Old English springan, spring or jump, which is from the Proto Germanic sprengan and Proto Indo European sprengh-, from spergh-, to move, hasten or spring.
Summer comes from the Old English sumor, which is just summer. It can be traced back through the Proto Germanic sumra- and Proto Indo European sem-, which also just means summer. So this one stayed pretty consistent through the years.
Ah, the season with two names. Which one do you use? Autumn showed up in the late fourteenth century as autumpne, from the Old French autumpne/automne and classical Latin autumnus, which is also autumn. Fall is much like spring, in that it came from what the plants seem to do. It appeared as another word for the season in the mid seventeenth century, a shortening of “fall of the leaf”. Because leaves fall, we call autumn fall.
Winter comes from the Old English winter, which means… winter. Not much change there. It’s from the Proto Germanic wintruz and is thought to mean literally “the wet season”. The Proto Indo European word for water/wet is wed-/wend-, the source for pretty much everything related to water. I guess snow is wet. I mostly try to avoid it.
Tony Jebson’s page on the Origins of Old English