Erm, cryogenics or botanics
A Human Biology degree is a good idea because that would cover everything. My university ran Genetics, Immunology, Neuroscience, Nutrition and Evolution all under that course, but due to going Major / Minor I had to skip out Neuroscience and Evolution.
Research for cloned brainless bodies isnt too far ahead, it is just that we currently cant even make organs, this is what area you would go into today in this field, trying to make organs. Immunology would cover transplantation along with immune system, bacterial / viral disease (everywhere will teach HIV extensively) and allergies, Genetics goes into stem cells, mutations, genes, babies, gender development, pre natal screening, genetic disease (personality + physical), cancer and cloning.
I think you would want to combine both of them like I did, Immunogenetics would be the ideal course for this, or a Human Biology one that covers genetics and immunology like mine did. You just need to make sure to avoid plain Biology though because you have to do boring plants, environments and ecsystems in that one.
Basically, life extension and stem cells is under genetics, transplantation is under immunology. You would want to focus in learning about DNA, telomeres and cell division for aging, but this is a very minor area and not fully understood, which means lots of room for reasearch.
Yea, I think it is immunogenetics you want, not Biotechnology as Biotech covers plants and insulin and unrelated things like that, immunogenetics covers cloning, DNA, Stem Cells and transplantation which are what you are wanting to go into.