I'd go with Eimear. Traditional names are very popular in Ireland and if she's being brought up there she will have to learn Irish and probably be taught the translation of her own name - it's simpler to go straight to the source spelling IMO (and I say that as the owner of a traditional Irish name, it's nice to see stories featuring your own name). I knew many Eimears and an Emer growing up. Lovely name.
I think either version would be perfectly fine in Ireland. Maybe for the sake of the non-Irish rellies, go with Emer? I knew loads of both Emers and Eimears growing up and know of some children with the name (again, spellings vary). It has lasted really well as a name.
A lot of Irish names that you encounter really only burst upon the Irish scene in the 50s and 60s but they have 'aged' much better than other 50s and 60s names like Linda, Lorraine, Tracey, Sharon, etc. I would say with the exception of Deirdre most Irish names are still popular in Ireland, and of course spreading to the UK too after Siobhan and Sinead made a huge breakthrough in the 70s and 80s.
I think it's a beautiful name. I had an Irish friend called Emer. She said it was a very old fashioned name,but she liked it. The only trouble was that people constantly pronounced her name incorrectly. Few people got it right. No matter how often they heard it. I'm in the U.K though. I'd imagine it wouldn't be an issue in Ireland.
I had to read the T�in B� C�ailnge (in translation) when I was at university in England. My southern lecturer and northern self were the only Irish people in the class. Every single English person pronounced 'Emer' as 'Emma'.
If you think your British relatives might do that, I would go with Eimear (even though I am more used to Emer, and prefer it).