Yes. Hello I hope that I can help a little.
Here goes
The name Birmingham seems to have come from England. The first mention of it is Beomra Birm is derived from Beorma (or Beornmund) — Beorma was probably a local Saxon tribal leader, 'ing' is derived from ingas meaning 'tribe of' or 'people of', and 'ham' is short for hamlet or "heim", village or homestead. Therefore 'Birmingham' roughly means "The home of the tribe or people of Beorma.
The Norman invasion of England ment that a lord was given the area around Birmingham and this name became the family name eg. Richard De Birmingham.
The first recorded Birmingham in Ireland was Piers De Birmingham. In Irish the name Birmingham is MacFheorais or son of Piers. The Birmingham's were granted lands in Louth, Kildare and Galway.
In Galway their strongholds were in Athenry and Dunmore where the remains of their castles still can be seen. This is the branch that I come from. The earliest mention of a Birmingham in Tuam is a bishop in the 1300's. From 1300 to 1800 the Birmingham family were quite powerful and owned lands in north Galway. One of the most famous was John Birmingham who discovered a crater on the moon which he named in honour of the family. They also had a vast estate in the east indies.
After this the family lost its power but a few Birmingham families are still in the area and are all spelled with an i.
I hope that this will help a little.