Yes, Shiloh and Solomon are very closely related names.
Genesis 49:10 reads, "The sceptre shall not depart from Judah, nor the ruler's staff from between his feet ... until Shiloh comes."
(I have combined the KJV with NIV here for effect.)
A related verse, 1 Chron. 5:2 (NIV), reads, "and though Judah was the strongest of his brothers and a ruler came from him, the rights of the firstborn belonged to Joseph."
The coming of Shiloh/Solomon is generally considered to have happened many generations after the time of Jacob. However, the above two verses taken together virtually prove that Shiloh followed immediately after Jacob.
First of all, Joseph was not considered to be a ruler but only "second to pharaoh." As shown in Archaeology & the Patriarchs, that unnamed pharaoh who appointed Joseph as Prime Minister was in fact his own half-brother Judah! Upon the death of Judah the birthright (kingly succession) then passed to Joseph. However, it was not Joseph himself but one of his sons, namely Shiloh, who sat on the throne upon the death of Jacob. Jacob and Shiloh son of Joseph had the same Egyptian name, Amenhotep, and their reigns were also entirely continuous. It should not be too surprising then that the two were combined as a single reign in the Kings/Chronicles narrative of Solomon.
Often even I tend to speak of Solomon as being only the pharaoh Amenhotep III, however strictly speaking Solomon incorporates the reigns of both Amenhotep II (Jacob) and Amenhotep III (Shiloh). As shown in A & P, Thutmose IV (Judah) predeceased his father Amenhotep II (Jacob) by a few years. Another crown prince was not appointed, however it was declared that the very young Amenhotep III (Shiloh) would be made pharaoh upon his death.
Forty years later, after the death of Amenhotep III (Shiloh/Solomon), the house of Yuya (Joseph) continued to hold the throne until the death of Tutankhamun. As stated in 1 Chron. 5:2, another ruler did come from Judah. Thutmose IV was the biological father of Aye, who became pharaoh after Tutankhamun. Upon Aye's death the great throne passed to the line of Reuben (Kings of Israel/Egypt). However, many other pharaohs (a.k.a. kings of Judah/Libya) were also of the natural line of Thutmose IV through Aye.
© Charles N. Pope, US Library of Congress. All rights reserved.