Simplest solution would be to count the number of friends each of your friends has in common with you, and invert that and use this as a measure of centrality.
Anything further than this is bound to require additional assumptions: you can always sum the distances (along the shortest friend path excluding you) between all your friends (and the one with the smallest "total distance" would then be "most central"). But then you have to decide: what is the "weight" of the longer distances: Say you are considering the distance from A and B to persons C, D and E, and these are respectively 3,1,2 and 4,1,1: do you consider the total distance the same?
Also, if you want to avoid totally disconnected people (which make the sum awkward, because you have to specify a hard number for "disconnected" people's distances), you will probably have to allow for connections outside your own friends circle (e.g.: you are friends with 100 people who know me, but you're not befriended with me, and the 100 people don't necessarily are friends either). But even then you may have disconnected nodes in your "friend graph".
Finally, you may also have to weigh the connections themselves: perhaps the date the friendship was established, perhaps the number of messages that have been posted on either's wall (which could even make the "distance" nonsymmetric), the person who initiated the friendship (sent the "request"), or details specified about the relationship (family relations etc) could matter for your "distance of interest".
All in all: you will have to specify what your goals are, and adapt you distance measure to it. There's bound to be quite a bit of literature about distances in graphs, but all will require figuring out which distance you are interested in.