Because this is something you’re interested in, right? So I’ll etymologize a bunch of organs. Not heart, though, because I already did it in an A-to-Z post. So on to the rest.
Lung showed up in the early fourteenth century, coming from the Old English lungen (which meant lung shockingly enough) and the Proto Germanic lungw. No, I have no idea how it was pronounced. Lungu? Lunge? Anyway, lungw literally translates to “the light organ”, and that’s light as in the opposite of heavy, not the opposite of dark. It comes from the Proto Indo European legwh (nope; no idea), which can mean not heavy or agile, and just happens to be the origin word for lever as well. Now the reason lungs are lights is probably because when lungs were put into water, they floated, unlike the other organs. Um, the lungs of slaughtered animals. I hope.
Liver comes from the Old English lifere, liver, and before that the Proto Germanic librn. The origins are kind of murky, but it might come from the Proto Indo European leip, which could mean adhere or fat. But that’s not certain. Are livers usually fatty? I’m not really up on anatomy.
Spleen showed up in the early fourteenth century and its origins aren’t Germanic but the usual Old French. It comes from the word esplen, which in turn comes from the classical Latin splen and Greek splen, both of which are just spleen. Spleen can be traced even further back to the Proto Indo European splegh, which sounds like the noise you make when you throw up but actually just means spleen. We’ve had a name for this particular organ for a long time.
Brain comes from the Old English braegen, brain, and Proto Germanic bragnam. There’s some contention about where it comes from before that. Some say it’s the Proto Indo European mregh-mno, which means skull or brain, but others say it’s the word bhragno, something broken. So it’s either a word that sounds nothing like it or a word that means nothing like it. Sure.
Stomach showed up in the late fourteenth century with the -ch end and in the early fourteenth century as stomak. Why they had to change it, I don’t know. Anyway, before that it’s the Old French stomaque/estomac, stomach, and classical Latin stomachus, which could mean stomach or gullet, or even taste or inclination. Latin of course took the word from Greek, stomachos, which had a similar meaning and was taken from stoma, mouth. Because the mouth leads to the stomach, the Greeks named stomach after it. Which stuck all the way through English, even if stoma didn’t.
Tony Jebson’s page on the Origins of Old English