It is possible to research anything. Just because something isn't true doesn't mean i cant research it. People research things like big foot and and other sorts of nonsense every day, the possibilities of stupidity are endless!. Reincarnation is no different. You may get horribly flawed information but hey, it is still research.
Almost anything, including reincarnation, can be researched, although this by no means guarantees anything significant will result from the research. To research reincarnation one could investigate religious views of reincarnation, interview those who believe in reincarnation, search for historical evidence of reincarnation, etc. There is much that could be researched, although it is doubtful any conclusive evidence for or against reincarnation could ever be found.
These days, and with the Internet, I think it is possible to research just about anything you can think of, including reincarnation. I am sure there are tons of books, stories, and theories out there on reincarnation that would help someones research and eventually lead to a very interesting research project.
It is possible to research reincarnation but I don't know if you will find anything useful or valid. You can research how the idea of reincarnation came about. You can talk to people who think they lived other lives. You probably won't be able to find out if it is true though.
I don't think that it will ever be possible to research reincarnation. The most we can do is rely upon anecdotal evidence for reincarnation, because there's no scientific way to actually measure whether or not it's happening. I guess you could say reincarnation is a matter of faith, not a matter of fact.
Even though some people believe they can be hypnotized into remembering past lives, I personally feel that there's too much power of suggestion involved when someone has been put in an hypnotic state. It would therefore not be any proof to me if someone said they had lived as other people in the past. There's no real way to support the theory, because people's stories are not a form of proof.