See the following question: Deleting duplicate rows from a table.
The adapted accepted answer from there (which is my answer, so no "theft" here...):
You can do it in a simple way assuming you have a unique ID field: you can delete all records that are the same except for the ID, but don't have "the minimum ID" for their name.
Example query:
DELETE FROM members
WHERE ID NOT IN
(
SELECT MIN(ID)
FROM members
GROUP BY name
)
In case you don't have a unique index, my recommendation is to simply add an auto-incremental unique index. Mainly because it's good design, but also because it will allow you to run the query above.