The Neanderthal is perhaps the most controversial extinct species eligible for cloning and resurrection, primarily due to logistics: The surrogate species would be us. As the most recently extinct member of the Homo genus, Neanderthals are widely considered a subspecies of modern humans. Cloning them might be controversial, but it could also be illuminating.
A Neanderthal clone would also probably be most viable. Scientists have already completed a rough draft of the Neanderthal genome, for instance.
The question is not so much “could we do this?” but “should we?”