Firstly, sorry for your loss. I hope I can help you. My mother is in a similar situation.
1. Write to your local Garda Superintendant. Inform him formally of your fathers death and request that the Garda "survivors Pension" is commenced at the same time as your fathers pension is stopped. The survivors pension is normally half of the members pension. Enclose a death certificate, a marriage certificate and a birth certificate for your mother. If at all possible , Put your fathers garda registration number on all correspondance.
2. Apply for the Widows Contributory Pension. This pension is based on either your mothers or your fathers PRSI contributions up to age 66.
If your father has been paying credits up to age 66, You have every chance of getting this pension in addition to the Garda Survivors Pension
You can get the relevant form from your local post office or download it from the Department website. Fill it out, send it off and see what comes back. Once again, you will need death certificate, birth certificate and marriage certificate
3. If your father was paying into the Garda Benevolent fund, then a 3000 euro grant may be fortcoming to cover burial expenses.
4. If your father and mother were members of the Medical aid health insurance, you will need to contact same and advise them to continue cover in your mothers name only
Best of luck with it.