Best way to migrate a huge SQL Server database with low downtime over network
Straight backup and restore is obviously out. I also wouldn't consider replication of any kind.

Database mirroring is relatively simple to set up, but requires real-time connectivity between the two servers, setting up of partners and endpoints, etc. Availability Groups could be an option, but on top of the networking complications you also have to have both servers as members of the same WSFC - which means they must both be in the same domain. This is not a typical setup (or could even be made to work temporarily) for a data center move.

My vote would be for log shipping. The nice thing about this is that you can use the backups and log backups you're already taking (right?) and don't necessarily have to have real-time connectivity between the two databases - they don't need to know about each other, you don't need to set up endpoints for mirroring, partners, security, etc. You just need a way to get files from the old server to a place where they can be restored on the new server. You can take a full backup well in advance, get it over to the new server, restore it, then apply (possibly diffs and) incremental log backups from that point right up to the moment of cut-over. The process is actually quite simple, and there are many tutorials on log shipping available online if you come across any difficulties.

If the web app is moving with the database, since DNS can take a while to propagate, you might want to make a switch in the connection strings of the old app to make them point at the IP of the new database server once it is writable, since - even after the switch, and even if your TTL settings are tight - clients may continue to hit the old web servers. It all depends on how much respect their providers give your TTLs.