Pretty common operation to migrate SQL Server from one cluster to another.
Generally it is advisable to build and test the new environment in advance, possibly even using replication and/or backup and restore with differential and tlogs to keep downtime to a very minimum when moving it.
There are a lot of little steps and i'm not going to document them, but one issue we recently ran into was an SPN on the old cluster was not "released / removed" from active directory when it was renamed and taken offline and online with the new name. So... when we tried to bring the name online the new cluster it would not come online.
And of course the error message was not nice and clear like "Hey stupid, there is already an SPN in active directory with this name", no it wasn't, so it took me a good hour to find.
Also don't forget the properties of the cluster (virtual name is in there) and to sp_dropserver sp_addserver after rename.
Thanks to Vipin for helping find this !