What was the given name of Rabbi Yose?
Was it Yosef or Yose? If it was Yose, is that a derivative of Yosef, or a totally different name with some other meaning or significance?
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I believe (though I haven't seen my proof quoted in any sefer1) that "יוסי" is a derivative of "יוסף."
See the Gemara Bava Metzia, 117a (last few narrow lines), where there is a discussion about a leak in a multi-story apartment building, whether the guy living on the bottom (who is getting damaged by the water) needs to repair it, or if the guy on top (from whom the water comes) needs to pay for the repair.
ר' חייא בר אבא אמר העליון מתקן ור' אלעי משום ר' חייא בר' יוסי אמר התחתון מתקן וסימן ויוסף הורד מצרימה
Rabbi Chiya bar Abba says the tenant on the upper floor fixes it, and Rabbi Ilai quotes Rabbi Chiya bar Yose who says that the tenant on the lower floor fixes it; and a mnemonic is "ויוסף הורד מצרימה, and Yosef went down to Egypt (Breishis 39:1)2"
2 The trick is that there are two Rabbis with the first name "Chiya," and the one whose father's name was "Yose" ruled that the tenant on the lower floor had to pay, and Yosef who went down to Egypt can remind you of this fact.