One name for use in a connection string
I'll expand my comment into a proper answer, as requested.

Since, in this situation, the desired result is more of an automatic redirection to the new server without needing to change connection strings then the following options are available:


DNS Alias (CNAME)
Hardware Load Balancer
SQL Alias
Local Host Record


Each of these has its' own merits, however since this change is environment wide you'll want to have the least amount of moving parts and administrative overhead.

A Local Host Record (editing the HOSTS file) would have to be completed on each server so that it could resolve the proper address. This would be bother administratively burdensome and also riddled with troubleshooting issues.

A SQL Alias would also work but would fall under the same set of issues as the Host Record.

Hardware Load Balancers (or software load balancers, depending on implementation) are a great alternative as they can do the DNS swap for you. This could bring up some issues with Kerberos or Certificate signing, however those application could be singled out and fixed (potentially, depending on your environment and resources).

The DNS Alias is roughly the same as the Hardware Load Balancer option and the same set of issues would occur. However this option does not require specialized hardware or software in order to complete.

I would recommend either the HLB or DNS Alias options.