Just to save us re-ploughing well-ploughed ground: have you searched the Clare Library forum at http://www.clarelibrary.ie
for family clusters of Mulvilles, Mulvihills, Mulvihils, Mulvehills, and all the other phonetic variants of the name ? There are lots of them there, even going back to to the Tithe Applotments in the early 19th century.
See, for example, Maria Mulville in the 1901 census of Knock Town at http://www.clarelibrary.ie/eolas/coclar ... 98_13b.htm
(we're not told when your Daisy Maria emigrated, but if it was after 1910 this may be relevant.....)
Also in the Clare Library collections, note that at http://www.clarelibrary.ie/eolas/guests2005.htm
Terry Murphy, of Queensland, Australia, was searching for Mulvihill ancestors from Kilrush, Co. Clare.
Once you have assembled a few 'suspect' families and localities you could have a shot at obtaining a copy of her birth certificate. You know the year of birth, assuming the information is accurate, so you are off to a good start. The registers of births were started in 1864 so she may be recorded. See How do I apply for a Birth/Death/Marriage Certificate or photocopy from the General Register Office?
Some births were not registered for one reason or another but, heck, no-one and/or no system is perfect.
And so on and so forth.