The 11 year old I *almost* understand, but not the 9 year old who I think is just trying to be like big brother. At the summer camp I know of, at age 11 kids start in on a leadership program. They do some activities with the younger kids (mostly as guidance), but they do a lot of career & college exploration. The 8th graders even get their paperwork done for the 21st century learning thing (basically, you outline what you're going to do in high school so that when you finish HS and you've stayed out legal problems, a chunk of your college is paid for by this program).
If mom chooses to let them stay home (and it is mom's choice -- she's the parent!!!), she needs to work with them to come up with a schedule of activities for them to do. Mondays -- volunteer at the nursing home; Tuesdays -- swimming and park; Wednesdays -- work in the yard, house chores; etc. Additionally, since many summer camps are now using activities to build on skills learned at school, what is mom going to do to keep summer learning from lapsing? Can she get them involved in a type of abbreviated homeschooling with some workbooks? Or can they be required to read 1 book a week? Can they use this time to focus in more on a subject area they truly enjoy but don't really get to study at school? I mean, if one of the kids loves bugs, but they only spend 1 week on bugs at school, he can now spend the whole summer getting really in depth on learning about bugs and visiting places that have different types on display, etc.
Mom also needs to needs to weigh the cost. At the summer camp I spoke of above, the kids pay a flat rate that includes lunch, snacks and field trips. Can mom save money by keeping them at home with a structured routine? Or does Mom feel like she doesn't have time, resources, whatever the case might be to give them the structure they need and therefore summer camp is an investment? There's a lot of factors to weigh -- not just what the kids want.