No one has any DNA that can be traced to a specific Neanderthal, no.
Of course people have DNA in common with their ancestors as a group, hell we share a lot in common with fish. But it's because these genes are common that they enter and re-enter your family tree at multiple points.
And your mitochondrial DNA definitely doesn't descend from Abraham, a male incapable of passing on mitochondrial DNA, so there's a reason I gloss over female line descent.