Edit December 29, 2015:
So I felt bad that I had not done enough research for this name or also that I really couldn't find very much. So I looked up each part of the name again. The latter part of the name which is Elisia means the following, Noble, Nobility, Kind, Friendly, From the Blessed Isles, and Devoted to God. So since I think those are great meanings for a part of the name I am going to add it above. I also found the pronunciation for Elisia so I added that above under Pronunciation.
So the first part Amph however was harder. I did read it was similar and came from Amphi like in Amphitheatre, if that is true then the first part means, On Both Sides, Around. I can't be for sure if that is right. Even if I am not 100% sure I am going to add those meanings above as I did for the latter part Elisia.
If the meanings are right then the meaning combinations could be, On Both Sides Noble, On Both Sides of Nobility, On Both Sides Friendly, On Both Sides From the Blessed Isles, ect.
So again I am not 100% sure these meanings are correctly applied to this name but from my research they are close and are meanings for each part of the name separated but not necessary correct when the parts are combined but they seem good to me. This is just one of those old tricky names whose meanings and History has all but disappeared with time and not being used.