Not sure where 'Tubermore' is supposed to be- but since you mention PRONI do you mean Tobermore in County Londonderry? PRONI will only keep records for Wills that were probated in Northern Ireland.
If this 'Annie' died in America then you need to find out where she died- each state keeps their own records and within each state the location of Wills vary greatly.
You will probably have to start tracing all possible branches of your mother's family to find an Annie and then see if any did go to America, etc.