I was wondering if anyone knows the etymology of this kanji 湯. I'm specifically looking for the Japanese variation of Bath or "yu". My wife's last name in kanji is 湯上 (Yugami).
Looking up the etymology of the Japanese kanji "湯", you will find "From Old Japanese. Used in the Man'yōshū, completed some time after 759 CE."
And looking up a Chinese etymology dictionary named 象形字典:
Basically translated as:
"昜" is the original word for "陽", meaning natural warmth. You can find the character "昜" in "太陽", meaning "the sun". And the radical "氵" represents "water", pronounced 「さんずい」, in kanji such as "泳" or "波". So if you compose "氵" with "昜", you got "hot spring".
As for the pronunciation, "yu". According to this site, it's not an on-reading(音読み), but a kun-reading(訓読み), so it's not related to Chinese pinyin. (By the way, the 音読み is トウ).
And according to 語源由来辞典:
The kun-reading(訓読み) is related to 「ゆるむ（緩む）」, which originates from "the relaxing feeling you get when you get into hot springs".
The meaning in modern Chinese is "soup" or "hot/boiling water". The simplified character is 汤. It's composed of a water radical 氵 and 昜, which means "to open out, to expand". The original meaning was "hot water, and the sound which emanates from it [熱水也從水昜聲]" — hence the usage of the 昜 component. (see 说文解字 at hanziyuan.net).
The modern Mandarin pronunciation of the 昜 component is (pinyin) yáng, which is somewhat similar to the Japanese pronunciation yu. I don't know if that's coincidental.
The page linked above also seems to say that 昜 is a phonetic component from "sundial" — which I can certainly imagine, with the sun 日 casting shadows 勿 on the sundial — but I don't see any indication of this character having that meaning in my Chinese dictionary.
You can also see the above linked website to see depictions of various script forms throughout history for that character.
As for the history of the pronunciation, the character's Chinese pronunciation hasn't changed much since middle Chinese (around the time that Japan became literate in Chinese characters). See Wiktionary entry for details. In pinyin, that's tāng, or /tʰɑŋ/ in IPA, so for the onyomi of tou, it's not a big leap to see how the velar nasal /ŋ/ opened up to a high back vowel /u/ when it crossed into Japanese. Or maybe that's just a long /o/, I'm not really an expert in Japanese (or anything).
Chinese/Japanese characters are one of the great things about east Asian cultures. There's a lot of history behind every one.