“Father-in-law” is 義理の父, giri no chichi. You likely know that 父 means “father.” But what about 義理? Turns out to mean “sense of duty, honor,” breaking down as duty + what is right and proper. Yikes! What a lot of pressure and dreariness!
Halpern notes that 義 can also mean “in-law, step-, and foster,” as in these compounds:
義父 (gifu: father-in-law, stepfather)
in-law, step- + father
義兄 (gikei: brother-in-law, stepbrother)
in-law, step- + older brother
義母 (gibo: mother-in-law, stepmother)
in-law, step- + mother
But the original meaning of 義 is “righteousness,” whereas “in-law, etc.” is the last meaning listed, so I strongly suspect that “in-law” sprang out of a sense of duty toward inherited relatives. That is, being nice to them is “what is right and proper,” no matter how one may truly feel!