It sounds very nice, I wouldn't mind to see dodos in real life.
If you look at pedigree dogs, certain breeds (the north american German Shepherd) are so disgustingly inbred it can be exceptionally difficult to get a healthy dog (I've seen a lot of breeders who've adopted dogs from Germany that have passed schutzhund so they can guarantee their dogs against hip displasia with some confidence). Then you look at another breed, like the Standard Schnauzer that it's very rare to get an unhealthy dog.
If you inbred a small group of NA German Shepherds it wouldn't take long before all puppies born would be virtually guaranteed to have displasia. If you inbred a small group of Standard Schnauzers it would take a long time before appreciable disease took hold of the line. From the AKA estimates only 1% of Standard Schnauzers have serious illnesses, which with controlled breeding could possibly be removed from a small group (if it even presented to begin with).
So a sufficient gene pool might not be necessary, supposing the animals that are cloned are healthy to begin with. Once they've been artificially bred past say 1000 members it might be possible to release the animals into the wild and have them successfully thrive biologically.
I suppose they could raise 1000 Dodo's in zoos or something (pigeon's are possibly closely related enough to give birth to and rear a Dodo) and then dump them all on Mauritius.
Most lower animals are ruled by instinct. A bird never 'learns' from his parents how to make a nest.
For interest sake, this was done for a while with a completely different species. Domesticated cows were implanted with fertilized Buffalo eggs (this was because Buffalo's carried Bovine TB and a TB free Buffalo is worth a lot more $$).