The Oxford English Dictionary, generally highly regarded, says this on the origin of "bother":



Of course, the OED editors don't speak much Irish. I'm reading an interesting article by Thomas F. O'Rahilly,. "Notes on Middle-Irish Pronunciation" in Hermathena, Vol. 20, No. 44 (1926), pp. 152-195. He argues p178 that bodhar is the Irish original, and it was borrowed by English (first into Irish English and then into English English) as "bother". He points out that first of all the OED miscites the Irish as "bódhar", when it should be "bodhar". Second, they claim that bodhar doesn't sound like bother, and so can't be the derivation. O'Rahilly shows in the article (from evidence like English forms of Irish placenames and names) that the Irish "dh" was pronounced like the "th" in "bother" well into the 1300s. So "bodhar" was indeed pronounced very similarly to the English "bother" into the 1300s, and is the derivation.

On the timing, he says that slender dh began to lose its dental character in the early 1100s and become confused with slender gh. But broad dh remained dental, and only gave way during the course of the 1200s. By 1300 most Irish people would no longer have had a dental dh, apart form a learned few, who may have retained it longer. He states that every manuscript after 1329 shows that dh and gh were then no longer differentiated. His timing for the loss of the voiced dental fricative is much later than other academics' of his time, but he had 44 pages of evidence in that article.