"Also, it states at the end of the letter she received that if the deceased has not left an estate/will, no further action is required. She is wondering if she should just ignore the letter. Is there anyway that the social welfare would find out that a will was made."
They are on to your friend. She needs to get proper advice. If there is money due from the estate to Social Welfare, that debt takes precedence over any of the assets.
Its a no brainer. They know he is deceased. A simple land registry search will probably show the house in the father's name. Even if there was no will, the monies due to Social Welfare, if any, still take precedence.
mf