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You would need to hack the registration form, and perhaps add some space for it in the db, to make a log the refer.
A possibly easier method would be looking into a profile module, something like that offered by Smart Factory, I'm not entirely sure as I haven't used this module, but I believe it allows custom fields, you'd have to try it and see though or ask on their support forums:
Smart Factory - Smart ProfileIf your intent on hacking the form yourself, then there are many posts within these forums which would offer some great assitance.