Henrietta Lacks Cell Line - Immortality
No, there was nothing special about Henrietta Lacks, apart from the fact that her cells have been used for so many years (without her knowledge or consent, by the way). She was not immortal, her tumor cells were. The HeLa cell line comes from her cervical cancer cells. The rest of the cells of her body were not immortal any more than yours or mine are.

Such immortality is a classic characteristic of cancer, in fact it is a hallmark of tumor cells. Normal cells die when they are told to, apoptosis (programmed cell death) is a normal part of a functioning, healthy organism. Tumors are collections of cells that replicate uncontrollably and do not do what they're told, including dying like good little soldiers. 

That is why we often use tumoral cells for cultures, since they have been immortalized, it is relatively easy to keep them alive in the lab. This means, essentially, that they produce an enzyme called telomerase which lengthens telomeres. In normal cells, the telomeres get shorter with each round of mitosis and when they get shorter than a certain threshold, they are targeted for apoptosis. Cancer cells are protected from this by the telomerase and can therefore undergo many more rounds of reproduction.

So, the cells are not special because of the person they came from, they are special because they are cancer cells.