[Bass4Gsus]

I think your best bet is to adopt the new name as the master file.

Then create a song with the "original" file name which simply contains text to remind you the name of the master song.

I play in 3 groups that have several common songs.
To avoid confusion, and to allow the groups to diverge, each group has its own copy of the file (i.e. 3 separate files).
To differentiate between these files, I append a short mnemonic to the end of each file name eg. " - MUJ", " - Pub"
I keep each groups files in their own collections.

Remember that a song can be in more than one collection i.e. you can put both master and "reminder" songs in both collections 

Cheers
Geoff

That's a great idea for this rare situation. I will reference the new file in the old copy and the old file in the new, just in case.  I often make copies if different groups have different arrangements that are different enough and add initials of the group to the song name for quick selecting.  Otherwise I would use the same song file.   I play with one individual who often makes radical chord changes etc and almost anything she touches I have to create a separate file, not just a copy to keep my sanity.

MobileSheets flexibility is amazing with collections, set lists, and even albums for fakebooks, sources and I use one field to identify who provided the music.  The most difficult was the initial scanning and cleaning up of the pdf's but now it's a breeze.  I love watching everyone else in the church group struggling with finding the music since it's 8 different songs each week, it only takes my 5 minutes to setup for the next week.  In other situations the lists are more static.