Migrating Standalone named instance to AlwaysON Scenario:
I have a named instance such as SERVER1\NAME.

I need to migrate this instance to an AlwaysON, but I'd like to use the same instance name in the listener. Is it possible?

I was planning a side-by-side migration:


Create 2 new instances. (SERVER2\NAME and SERVER3\NAME)
Create a Listener called SERVER1_NEW.
Once everything is ready for the cut-over, just rename the SERVER1 to SERVER1_old and then rename SERVER1_NEW to SERVER1.


Is that going to work? 
The plan is to avoid changing the connection strings everywhere.
Yes, its very much possible and I completely concur with your idea as starting with new server and do all the required and once all set-up, get a short window of down-time. Take database backup and restore it on the new server. Once sanity and basic testing is completed successfully, switch off the old server or rename it with some other name and allocate that name to new server.


Create 2 new instances. (SERVER2\NAME and SERVER3\NAME)
Create a Listener called SERVER1_NEW. 
Once everything is really for the cut-over, just rename the SERVER1
to SERVER1_old and then rename SERVER1_NEW to SERVER1.


You may read wonderful article on how to do set up of windows always on availability group by Priyanka Devre as below:

https://www.sqlshack.com/configuring-a-sql-server-alwayson-high-availability-group/

Hope above helps.