The problem with finding the origin of a folk belief like this one - that cat's have nine lives - is that when there is no clear answer, folklore typically invents one (the folk abhor a vacuum!). The idea that Ancient Egypt is the source of this belief is likely a bit of folklore in itself, caused at least in part by the understanding that ancient Egyptians revered the cat and are well known for their fascination with the afterlife.

This site attempts to answer the question presented here, but it has unsatisfactory explanations. It attributes the Egyptian source to the Encyclopedia of the Unusual and Unexplained: Superstitions, Strange Customs, Taboos, and Urban Legends - a site that does not inspire trust. Of use with the first site, however, is the observation that Herodotus, who is a contemporary commentator on Egyptians and cats fails to note anything about the multiple lives of cats: if this was such an important attribute of belief that we still remember it, one would assume the ancient Greek historian would have mentioned it. The site also notes the early reference to the cat's nine lives in Shakespeare's Romeo and Juliet, but that does little more than pin down an early occurrence - not that still earlier ones don't exist.

The Wiki site on cats lends a bit of insight even though we all know that source is to be shunned. It suggests that "In many countries, they are believed to have nine lives, but in Italy, Germany, Greece, Brazil and some Spanish-speaking regions, they are said to have seven lives, while in Turkish and Arabic traditions, the number of lives is six." We can't trust this source, but it raises a good question: if other cultures - particularly other European cultures have numbers other than nine for the life of the cat, we can assume that the folk tradition is a matter of multiple lives with no consensus on the exact number. In that case, we can set aside the idea that the number nine as significant and focus instead on the serial resurrection of the cat. We could return to the Egyptians as the source of that concept, but we would certainly need evidence rather than conjecture.

Katharine Briggs, the esteemed British folklorist has a book titled Nine Lives: The Folklore of Cats. She was big on description and short on explanation. I don't have the book, but I suspect it won't explain this, and I suspect that your question can only be left unanswered.