Visas: travel purpose for a family member traveling with a student to Ireland

Assuming that the only purpose that your mother has to visit Ireland is to be with you, the only appropriate visa is a tourist visa, which covers visits to family and friends. Your mother is not dependent if you, and therefore can not apply for a dependent visa in the same way that your spouse or child could. By filling out the forms visiting the family, it would be an appropriate reason to travel.

You must definitely wait until your student visa is approved before applying for your mother's visa.