teeeechnically they got their name not from the william blake quote, but from the aldous huxley book about mescaline "the doors of perception" which got its title from the blake quote.
Good point. Aldous Huxley's book was probably the main inspiration for their name as you point out. I used Blake as the answer to the question because ultimately the phrase "the doors of perception" comes from him. I also believe Morrison did read quite a bit of Blake as their name is not the only reference to his poetry in the Doors music, and so it is possible he took the idea from both Blake and Huxley. I guess it's up for debate. Perhaps I may change the question a bit. Thanks for the comment anyway.
Tried Huxley like six times even though I know he wasn't a poet. I was sure the name came from him. Didn't know the phrase was originally form William Blake.
Nominated ;-)