I've played both games to completion, and it was the right call to make, even if the mother didn't know why she was making it. Here are a few of the objectionable things featured in each game that should put up red flags for a parent of a small child:
Red Dead: Gore/shooting, fairly tame torture, cursing, prostitution & mistreatment of women, race issues, a graphic (by video game standards) and unnecessary sex scene, theft, alcoholism, necrophilia and arguably terrorism (depending on how you see the actions you take in Mexico), killing & skinning animals.
Black Ops: Gore/shooting, fairly tame torture, cursing.
Black Ops really is a much more tame game than Red Dead. Red Dead just arguably has more artistic value, and I don't think an 11 year old would particularly care about or even understand that.