Erin Go Bragh!
I vaguely remember seeing some debate about the origins of this phrase on the GAELIC-L list about this, but haven't checked back there in a (long) while. Anyway, the word Erin is the anglicisation of the Irish word Éireann which of course translates to Ireland. Whether this refers to the Island or the Country is a political point that I won't pursue. The remainder of the phrase is more of a challenge. There is no "bragh" in the Irish language so it's somewhat difficult to figure out from what original Irish word this came. Among the possibilities are:
Brach; unlikely as this means pus
Brách; more likely, as go brách means for ever, though with a negative it can mean, for example: Is fearr go mall ná go brách (Better late than never).
Braich; Hmm... could be, as this means Malt (!)
Bráth; possibly, though I'm not convinced as Lá an Bhrátha translates as Judgement Day.
Breá; I personally think this is quite likely, as it translates as good or fine, for example: Tá Sí go breá means She is fine.
Bréag; I don't think so; this means false or a lie, example: fiacla bréige would translate as false teeth