From what I know, eHarmony asks you a long list of questions, and then tries to match you up with people they think you would be a good romantic match with.
Not sure how well the above works, but I always thought a similar thing that applied to finding people whom you would click as friends would find many applications:
Moving to a new city, moving into a new apartment complex, starting a new job, starting college, going on a cruise with a ship full of people you don't know, etc.
I guess there are websites that help you expand your network of friends, but as far as I know, no website runs a test that is analogous to the one on eHarmony.
My questions are:
1) Do you think that a questionnaire + algorithm could ever be developed to find people that you would click with as friends, or are human nature and human interactions too complex and unpredictable?
2) Assuming such an algorithm is possible to develop, would it get traction, would people use it, would people pay for it?
2) Not easily. The biggest problem is that the people who most need help finding friends are the ones you least want to find on a friend-finding site. Every dating site seems to be drowning in this problem, which means they have to resort to scam techniques and exploiting new, naive users as much as possible. OkCupid is the only exception that I've seen. Hard to say whether you can replicate what they're doing (that is, not sucking) in the Platonic realm.