From your description it sounds like you have added the conditional field AFTER the original merge field instead of REPLACING it.
«Request_Staff_Person»{IF{MERGEFIELD Request_Staff_Person}= "Mickey Mouse" "Minnie Mouse"}
The field goes INSIDE the conditional field i.e.
{IF «Request_Staff_Person»= "Mickey Mouse" "Minnie Mouse"}
or with field codes toggled (ALT+F9)
{IF{MERGEFIELD Request_Staff_Person}= "Mickey Mouse" "Minnie Mouse"}
In the case of the latter all bracket pairs are inserted with CTRL+F9
Or you can insert it from the mailmerge tab of the ribbon (Rules)