Many variations of the name exists e.g. Myer, Meyr, Meyer, Meyerson, Meier, Meir, Maier, Maiorana, Mayer, Mair, Mayr, Von Meyer
The name is documented in England since the Doomsday book and considered Anglo-Saxon and could have spread into Ireland.
If you want to research Annie specifically I think you may need to consider as Rosie99 suggested start with an official record such as her marriage certificate. This will place her and give her father's name. If she wasn't living at home it may give her father's address especially if he was living.
In light of your name, her name and her approx DoB is this her marraige?
John DORAN & Annie MYERS Feb 4 1883 in Kildysert -
LINKThe name is registered a few time in this district -
LINK