Why did Moses’ father-in-law have three names?
If “chosen” means “in-law”, and not specifically “father-in-law”, there’s no need to assume that one person had three different names. Reuel was priest of Midian when Moses arrived, and Moses married his daughter, but Yeser was head of the family when Moses asked his permission to return to Egypt. Yisro (a variant spelling of Yeser) brought Moses’ wife and children to him at Mount Sinai; by that time he was priest of Midian. Hovav, (another) son of Reuel, stayed for a while, and Moses asked him to guide the Jews in the desert, which he apparently declined to do. On the other hand, it’s perfectly possible that Reuel was also called Yeser/Yisro; we know of several people who had two names, including Sarah/Yiscah, Yaakov/Yisroel, Esau’s wives, and many of Jacob’s grandchildren.
All the Best,
Rabbi Azriel Shreiber