Probably call him Hughie.

That information, that Hui is a nickname from Amenhoptep I've got from (if I remember it right, often I don't know where I've got some info from). Yes, it's there:

Sometimes a nickname is not so easy to explain for example Huy or Huya is used for the name Amenhotep, much like today Beau is used for Beauregard.

Hmmmmm....I don't really see the association between Amenhotep and Huy in comparison to Beauregard and Beau. Obviously Beau is part of the name Beauregard, but Huy is not a part of the name Amenhotep. You've got me quite curious about this name now. I'm going to do more research on it.

Yes, because Min was associated with fertility and it was Khonsu who was the major lunar deity. Thoth holds the sun upon his head does he not? He was the god of scribes and wisdom however, a far cry from fertility or the moon. Min is not associated with the moon however and Khonsu is not associated with fertility. The only link is the evolution of the dieites Min and Amun to form Amun-Min, as Amun was the "father" of khonsu it would provide a link. Otherwise I am baffled.