My grandchild live with me until Aug. 10th, I know his mother will claim him, but do I have the right to claim him, since he lived with me for 8 months?

The child cannot be claimed on more than one tax return.  If your grandchild lived with you for more than half the year---and you say he was with you until August 10--then you are the one who can claim the child as a dependent.  The IRS cares about who the child lived with for the most nights.   If he was with you for at least 183 nights then you have been the custodial parent who can claim the child.   If someone else files a 2024 tax return before you file, and claims the child, your own e-file will be rejected.   If that happens, then you will need to file your return by mail.  The IRS will sort out which of you can claim the child.  Be prepared to show (at a later time) that the child lived with you.   It would certainly be best if you and the child's mother can have an understanding before tax time of who can claim the child to avoid an unpleasant and contentious situation.