Effects of renaming FC and using its original name for AG network name We have a Failover Cluster, named MainSQL consisting of MainSQL01 and MailSQL02 which are default named instances and we can connect to whichever is the active node with just simply MainSQL. 

If we were to rename the FC to MainSQL-NY then the connection would then be MainSQL-NY\MainSQL as far as I can tell, since the instance cannot be renamed. If we then put this cluster into an Availability Group in a DR site with another cluster (let's say  MainSQL-NJ\MainSQL with its own two nodes) would we be able to re-use the MainSQL name when creating the AG Virtual Network Name, thus allowing connections the AG and its underlying active cluster node to still be made simply using MainSQL as it would once again be MainSQL\MainSQL?

The alternative is to rebuild the clusters or just force all our applications to update their connection strings, but I'm curious if this is feasible.
An alternative you might not have considered is DNS. Obviously changing the connection string is easiest, but if that is a problem for some reason, you could leave the connection strings alone and remap using DNS. More importantly, think about these questions:


Why do you need to rename the server/cluster? 
Is the rename essential to business processes, or is it just a "tidying up" kind of thing?
Do you really need the geographical location in the name?
Will it really make sense to refer to it that way, since it seems logical
that part of the HA/DR topology might be to fail over to a different
geographical location?


In my experience most people have left that stuff alone, and documented very well that the original primary server was not renamed when they expanded geographically (or wished to rename for other reasons). You could also leave it the same and use DNS or the hosts file to let the applications and other servers/locations think the server was renamed, when really it wasn't. 

The actual rename is the hard part IMHO, and I would be looking at any alternative I could to avoid doing that. DNS, hosts files on all relevant machines, aliases, and changing connection strings are all ways you could bypass the actual physical rename.