I feel like I've been reading about plans to reintroduce woolly mammoths for several years now. This link doesn't mention much in way of timetables or specifics. Is this turning into nuclear fusion or are we really making progress on this? What are the odds I'll get to see a live woolly mammoth in my lifetime?
One good reason is that existing species are in some danger of collapsing from the extinction of keystone species.
A good example are joshua trees. As I understand it, giant sloths distributed joshua tree seeds. (i.e. they ate the huge seeds, wandered over the next hill, and crapped them out in a big pile of fertilizer). Without giant sloths, joshua trees can't migrated outside of their existing habitats.
That's all well and good while the climate isn't changing dramatically, but we're not in that world any more. The current estimate is that unless some substitute animal can be found, joshua trees with be extinct within the next few hundred years.
So restoring ground sloths would save an existing species from extinction.