Der Spiegel interviews a Harvard professor who thinks we're nearing the point where we could implant Neanderthal DNA in a human egg. Of course, that would take a woman willing to carry a Neanderthal to term. Actually, several women, because what's the point of growing just one?
No, you would certainly have to create a cohort, so they would have some sense of identity. They could maybe even create a new neo-Neanderthal culture and become a political force.