Billy Mailman's answer is rather complete, but just simplifying things, the following worked well for me:

If you are playing on a computer:

 1. Open the console
 2. Right click on Serana **OR** type `prid 02002b74` and hit enter to select Serana
 3. Type `Setpqv DLC1NPCMentalModel Lockedin_var False`
 4. Hit enter key

This will cause the "Follow me" dialogue option to reappear.

This solution was found on [this link][1].


  [1]: http://www.uesp.net/wiki/Skyrim_talk:Serana