But why is there a chirik under the bet in bin-Nun? It should be a segol, just as by everyone else's name?! There is surely a real answer somewhere, but the cute answer, cited from Mishulchan Gavoah, is as follows. We know from a midrash that the extra yud tacked on to Yehoshua's name was taken from Sarai, who became Sarah. But there is a sheva under that yud. Where did it come from? The answer is that there were three dots in the segol in the word "ben" and two of those dots were taken to form the sheva, leaving only one dot, for the chirik.
|טז אֵלֶּה שְׁמוֹת הָאֲנָשִׁים, אֲשֶׁר-שָׁלַח מֹשֶׁה לָתוּר אֶת-הָאָרֶץ; וַיִּקְרָא מֹשֶׁה לְהוֹשֵׁעַ בִּן-נוּן, יְהוֹשֻׁעַ.||16 These are the names of the men that Moses sent to spy out the land. And Moses called Hoshea the son of Nun Joshua.|
I think (and hope) this explanation was intended as a joke, and as an exercise in creativity. Otherwise, there are obvious ways of debunking it. As one fellow at my table pointed out, Hoshea bin Nun, without the yud, still has a chirik. I could add that the orthography of the sheva did not exist at that time. Also, the tzeirei under the shin becomes a kubutz, so where does that extra dot come from? Still, very cute.