What was the name of the Mills Brothers recording that had the lyrics 'I wish I knew the name of the girl in my dreams so I could change her name to mine'?

Wasn't aware of a Mills Brother's recording, but it sounds like you're referring to the song Dinah Washington recorded as "I Wish I Knew the Name of the Boy" ... I wish I knew the name of the boy of my dreams, so I could say his name is mine ... I know his tender lips, I know his sweet caress, but gosh darn it all, I don't know his address ... That sound like the right song?

I believe the song you're looking for is "I Wish I Knew The Name (Of the Girl Of My Dreams". Released on Decca in 1947, the flip side was "Manana". Dinah Shore may have done a version of it herself, but the Mills Brothers version is definitely the 1947 one you're looking for. The song is I Wish, recorded by the Mills Brothers in 1944 (?) Hear it on http://www.jazz-on-line.com/a/mp3o/DEC72643.mp3