Hi
You might like to have a look at thhe Leitrim-Roscommon site -
The 1901 census for some counties have been put on here.
When you find the 1901 search page -
Select county Mayo and just put Tunney in the 'Head of Household surname'.
Then select county Mayo again but just put Tunney in 'Other Enumerators in Household' i.e. living in someone else's house.
Hopefully you should have all the Tunneys in Mayo in 1901 - and you may get some clues - Maeve might have been a pet name as I couldn't see anyone of that name.
I believe the 1901 census of Ireland is due to go on line early next year.
http://www.leitrim-roscommon.com/index.shtmlHope this helps and Good Luck.
Regards
Neptune