First off, I recommend separating the first and last name into separate columns. (first_name and last_name)
If that's the case, then the solution is much easier. You can update the field in mysql using this sql query:
UPDATE users SET full_name = CONCAT(UCASE(SUBSTRING(full_name, 1, 1)),LCASE(SUBSTRING(full_name, 2)));
However, this will make the result be:
MONICA Brown ---> Monica brown
DOnald Doe ---> Donald doe
In order to capitalize the first letter of each word, you will need to create a mysql function.
Here's a site with an function you can probably use:
http://www.thingy-ma-jig.co.uk/blog/30-09-2010/mysql-how-upper-case-words