You'd need to implement a lookup table with the alternate names in it:
Base | Alternate
----------------
Robert | Bob
Elizabeth | Liz
Elizabeth | Lizzy
Elizabeth | Beth
Then search the database for the base name and all alternates. You'll end up with a number of multiple matches which will then need to be checked to see if they really match based on a comparison of whatever other data you have in the two databases. Maybe the dates of the records in each database could be used - records entered close in time indicate the same person.