You haven't said where your child and grandchild live. That is key.
My first comment is that if your daughter claims her son as a dependent, then you can't claim your daughter, regardless of any other factors we discuss.
My second comment is that you are probably allowed to claim your daughter as a dependent, even if she doesn't live with you, but we need to know more. If she does live with you, you are probably also allowed to claim the grandson as your dependent, but only if the grandson lives with you in your home.
My third comment is that the father can't claim the grandson as a dependent unless either (1) the grandson lives in the same home as his father, or (2) your daughter signs a waiver.
My fourth comment is that, assuming that you are allowed to claim your daughter and grandson as dependents, you and your daughter would have to work together to see which method gives the largest refund or lowest tax. You could claim a $500 dependent credit for your daughter and up to $2000 child tax credit for the grandson. Your daughter, depending on how much she works, might be able to claim up to $1400 child tax credit for the child plus Earned Income Credit, which might be more than you would get. It depends on your taxable income, your daughters' taxable income, whether you work or are retired, and other factors. The only way to know for sure is to test it both ways.
For a more specific answer, we need to know where your daughter and granddaughter physically live, and and the approximate income earned from working by both you and your daughter (not counting pension, IRA, social security, investments, etc.)
