I'm pretty sure I read this in Analog, but it's possible it was in one of the other magazines, like Asimov's, probably in the 1980s or 1990s. I have a vague recollection of a black-and-white drawing of a jungle with a bird.
I don't really recall much beyond what's in the title. Researchers have, by techniques I no longer recall, reconstructed many extinct species in the pre-human Hawai'ian island ecosystem. They have bred these species and tried to reconstruct the ecosystem itself. (I don't remember if they had taken over one of the islands, if they had a preserve on part of an island, or if they were doing this in some kind of containment structure, like a dome.)
One of the last species to be reconstructed and released is a bird that is somehow special; I believe it was mentioned in Hawai'ian traditional narratives (pre-colonization) or something like that. The bird or birds (there must have been more than one, right?) are released, and gradually the researchers start to feel the mental presence of some kind of awareness that seems to be emergent from the ecosystem.
I don't really recall how it ends, but the researchers may have been succumbing to pressure to give up the trappings of civilization and go live wild in the restored jungle.