A very neat idea. I could certainly relate to the protagonist, which goes a long way toward me liking a story. I liked the way that the circumstances set the neighbor up as his successor in the Bird tending business. She even gets to practice a little bit before it's hist turn to go.
I think I would've liked the story better if there had been some attempt to change the status quo. Apparently this is the way the world has always been, and will continue to be, so there wasn't a great deal of tension for me. It seems that there's enough evidence that this system works, that reading the name is tied with the moment of death. But since there was no challenging of the system in the story, my mind is more occupied with testing how the system works, by finding out what happens if:
--A person is never named.
--A person is named, but their name is not given to a Mori bird.
--The name slip is burned before being read
--Someone illiterate is the only one to see the name
--The wrong name is written on a slip
As it was, it was a well-written bittersweet little tale about death and unrequited love, but I don't think it will linger in my mind as long as it would have if there had been something to raise the tension.