The Torah refers to Yehoshua as Yehoshua Bin Nun. (See here for an example).
Why Bin Nun and not Ben Nun?
Are there any places in Tanach where he is referred to as Ben Nun?
Is there anyone else in Tanach who is referred to as Bin instead of Ben?
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According to Radak (Yehoshua 1:1), this is grammatically proper for "ben" to become "bin" when it and the following word are small and connected together in speech.
Other examples where "ben" becomes "bin":
Yehoshua is always referred to in Tanach as Bin Nun. In Nechemia 8:17 he is referred to as יֵשׁוּעַ בִּן-נוּן.
The only other case in Tanach where a person is called Bin is in Mishlei 30:1 דִּבְרֵי אָגוּר בִּן יָקֶה. However Rashi cites the Midrash that Shlomo is called Agur, meaning the one who gathered this information, "Bin" meaning not son here but that Solomon understood this wisdom, and Yakeh because he then "spit it out" for others.
The Chasam Sofer in Toras Moshe says that since the added Yud needed a Sheva underneath it the two dots were taken away from Ben and made it into Bin.
The Ramban Shemos 33:11 says that Yehoshua was known as Bin Nun to show him honor, as the word Bin Nun come from the word Navon - to show that there was no one greater than him in Chochmo and understanding.
See this link for additional reasons http://www.chabad.org/library/article_cdo/aid/533436/jewish/Why-is-Joshua-referred-to-as-bin-Nun.htm
Speaking purely linguistically, Semitic languages originally had only 3 vowels: a, u, i which is still the case in standard Arabic. Again, speaking purely linguistically, Torah has many examples of grammar and words more ancient that most of its text. 'Bin' may be an example of such older pronunciation that remained in his family, or in the tribe of Ephraim. Or the specific phonetic environment preserved 'i' in 'binnun', 'binyake', 'binyamin'