Earlier versions of MLS allowed more than one member to be associated with an MRN. See this post
, for example.
That fact makes this difficult to sort out. You might have three different users associated with John Doe, and you can't make any change to any of those MLS users as long as that situation exists. So the only solution I've come up with (and yes, this makes me uncomfortable, but I haven't heard of any other effective solution) is to associate one or more of these MLS users with a member who is unlikely to ever use MLS.
For example, you may find that John Doe is associated with an MLS user called "Elders Quorum President" (a calling he once had) and with "Bishopric 1st Counselor" (the calling he now has). Rename "Elders Quorum President" to "ZZ - EQP" and associate it with Sally Jones or some other member who probably won't ever have an MLS calling. The "ZZ - " scheme is designed to move all these to the end of the list and make it so no one is tempted to login with those; you can change the password to these outdated accounts while you're at it.
Keep careful notes of what you do, and work through every MLS user. Removing the duplicate associations for these obsolete users should allow you to clean up all the current users to have appropriate names.