In that case you should be able to rename the server following bringing it back online. I have done this and it does work, even for the distributor.
The big caveat with having the distributor set up for DR is that you have to have the LSN's from the publisher transaction log in sync with what the distributor has. Unless you have both the distributor and the publisher mirror at the exact same point you will end up in an out of sync condition and will end up either reintializing OR ignoring the commands that have not been sent from the transaction log to the distributor to get replication flowing again, which results in an out of sync condition.
If you use storage level replication, and you can snapshot both the publisher and the distributor volumes at the same time you should be able to eliminate that. Not a cheap solution of course.
Anyway, my point in bringing that up is that for DR of the distributor your cheapest alternative may just be to have a server set up (but powered down to save licensing costs) to the same exact version and patches, and then just re-establish replication through scripts in the event you actually had a DR scenario. The coverage provided isn't any less than what you will likely get with the mirror. You could also rebuild the replication without initializing and do a manual sync as well.
There is a good bit to this so I'll leave it here. If you have specific questions about what I stated above, let me know.
“He is no fool who gives what he cannot keep to gain that which he cannot lose” - Jim Elliot