My daughter and her son lived with me for the entire year without paying rent or helping with the utilities. She did provide the majority of food for herself and her child as well as most of his clothing.
I feel I have the right to claim him as I provided the vast majority of support. What I offered to do was compute how much I would save if I claimed him, then let her claim him (because she would get more than I would) and have her pay me what I would have gotten.
Once I told her how much the credit is worth to her, she wants to claim him and is being quite the brat about it.
Is it her right to claim her child even if they freeloaded off me all year?

A grandchild can be a dependent if he meets other IRS requirements. He must be under age 19 or, if a full-time student, under 24. He has to live with you for 50 percent of the year and provide not more than half of his own support. There can not be another person who can claim your grandchild as a dependent. For instance, if his mother lives with you and she provides his support, she gets to claim him and you don't

"and she provides his support" is the crux. She is not providing a majority of the support when she pays no rent or utilities. I called the IRS. The first big point is: 1) Can I claim my daughter as a dependent? Because If I can, then I can claim my grandson too. My daughter is 23 and not a student so the answer is no. 2) If she claims him, I can't. Done. Over. The IRS won't look at who paid what or who contributed more. While I can claim him with her permission, if she doesn't agree she will win any tiebreaker decision.
I don't think this is fair to grandparents in this extended family environment, but that's what they said.

When she pays no rent or utilities, she obviously isn't providing the majority of support.