I realise this thread is old, but I said I'd bump it on the off chance before pming the OP.
Just googling my own surname brought me here, unfortunately I'm not much into geneaology
Rabbitt is very much a Galway name, google Rabbitt's pub if you're interested in that aspect.
Family legend has it that some time in the middle of the 19th Century my Great something Grandad emigrated to California from somewhere near Tuam (county Galway) leaving behind a rake of brothers and sisters.
He struck it lucky in the gold rush and came back, only to find that the rest of the family had all emigrated to various parts of the world, and there were no forwarding details.
We are erm, fertile folk however.
Probably too late for you but the 1901 and 1911 Irish Census is online -
http://www.census.nationalarchives.ie/The bit about anglicanization of the surname is also a family story I've heard. It was originally a name that sounded like the Irish word for Rabbitt - something like Coinín (pronounced Cuneen)
In terms of record digitization, it's worth searching for Rabbett too. My family's 1911 census comes up as Rabbett when searched, even though the scanned form shows the dot, the looped handwriting makes it look like an e.