First of all, IGI has some, not all, Irish births 1864-1882 so it's not surprising you didn't see any births listed. Ballymena Library has the 1901 Census for Co, Antrim and Coleraine Library has microfilmed copies of some, not all, church records.
The index to Irish civil registrations is online at LDS Pilot site with only 3 records for Rachel Hamilton listed- and none of those close to the details you've given. Perhaps she was registered without a Christian name or not born in Ireland. Have you found the family in 1901 Scottish census to see what birthplaces are given for Rachel and her siblings?