Hello - native Memphian here. Memphis was founded in the early 19th century by several well-known figures, including Andrew Jackson. There's debate as to whether Jackson himself or James Winchester was the one to suggest 'Memphis'...but in truth, that detail's probably lost to history. What we do know, though, is that this name in particular sort of suggested itself because of the similarity of Memphis, TN's position on the Mississippi River to how Memphis, Egypt is also situated on the Nile.

[EDIT:] Some ambiguous wording in my last sentence there. I meant that the similarity is only that they are two cities on two major rivers - not the actual position relative to the rivers.

I actually addressed this question in a paper on Egyptian Revival architecture in the United States! Many settlers in the US drew parallels between the Mississippi and the Nile. The Mississippi was sometimes referred to as the Nile of the New World. So, some settlers in the Mississippi Valley decided to name their city after cities in Egypt. There's also a Cairo, Illinois, which is located right where the Ohio River meets the Mississippi.