SSI is needs-based. You are supposed to report to the social security administration, what family resources are available to support the child. You are supposed to report this regardless of how you file your tax return. It is possible that if your mother claims the child as a dependent, this might trigger a review of the situation, but I can't tell you for sure. Since you are supposed to be reporting your mother's support anyway (if you all live together in the same home), putting her on the tax return won't change anything. But if you have not disclosed your full situation, it might. I have no guaranteed answer.
Ignoring SSI, the tax code says that, if you, your daughter, and your mother (the grandmother) all live in a home together, your mother (the grandmother) can claim the child as a dependent if,
1. the child lives in the same home as the grandmother for more than half the nights of the year, and
2. the grandmother's income is higher than your income (grandmother's income is higher than the parent's income).
The fact that your daughter receives SSI does not change this calculation.
