My mother received SSI and considered disable due to her mental health. She died at age of 44 as a result of treatment of her illness. She had 9 children at the time of her death ranging in age 13 to 27. None of the children every received any benefits. My question is under what circumstances were they or were they not entitled to SS benefits?
If by SSI you're referring to Supplemental Security Income and if those benefits were the only benefits that your mother received, then it's likely that no survivor benefits would be payable. SSI is a needs based benefit payable only to the eligible individual, and there are no auxiliary or survivor benefits payable from SSI.
It's possible that there could be survivor benefits payable from your mother's Social Security record, but only if she was insured for Social Security benefits. If what she received was Social Security disability (SSDI) benefits, that means that she was insured and survivor benefits could potentially be payable. Children of a deceased worker can potentially be eligible for benefits until they reach age 18, or age 19 if still in high school, or at any age if they became disabled before age 22. An application for benefits is required, however, and you normally can't claim survivor benefits for more than 6 months retroactively.