I know you were looking for a specific name to put with Put in Egypt. There were two queens of the Old Kingdom named Iput. The first was a "King's Daughter", "King's Wife", "King's Mother", and "Great of Scepter". There was also a "King's Son" in the very early Old Kingdom named Menehpet. An early "King's Daughter" was called Shepsetipet. (Ipet meant "festival", but was associated with the innundation, so possibly there is a connection with Japheth/Put.)
But, honestly, the exact name or names of Japheth/Put are not clear. He could correspond to Weneg or Sened or one of the other early pharaohs we know little about. Or he may not have held a kingly/pharaonic status at all in Egypt.
Later "Josephs" were renowned more as viziers than kings or pharaohs in Egypt. We do have much better insight into these later incarnations of Ptah. For example, the 12th/13th Dynasty (Middle Kingdom) Joseph was called Inyotef (IV) and Ush-Piye, and was a deified prime minister. The 18th Dynasty Joseph was called Yuya, also a leading vizier. But he was not a pharaoh and some don't even think he was royal. The 22nd Dynasty Joseph was called Iuput. He was High Priest of Amun and Governor of Upper Egypt, but not specifically a pharaoh. There were however two Iuput's of the 23rd Dynasty who were pharaohs. The 25th (Nubian) Dynasty Joseph was called Piye and Pi-ankhy. He in fact did rise far above the station of vizier, but this was more due to his usurping/adding to his resume many other major roles.
In short, a pure Joseph figure was not expected to become a pharaoh, and quite possibly because the first Joseph after the Flood (Biblical Japheth/Put) had not become a pharaoh. Nevertheless, a prince in the role of Joseph would be expected to father one or more sons who did become pharaohs and establish a lasting royal dynasty.
Note that all of the known Joseph figures after the early Old Kingdom had names that referenced Yo/Yu/Ye/Ya, forms of Ptah. The Egyptian name of Japheth quite probably did as well, but it isn't possible to positively identify him.
Sorry, that's the best we can do right now.
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