Thanks for the description of the kids level (below average).
I actually think their time would be better spent on either doing remediation or advancement in their core topics or in just some useful use of the summer (archery, building stuff, whatever). This is NOT to discourage you, though.
My advice would be to keep things light and fun. NOT rigor city. Brain teasers, properties of numbers. Polya (and maybe even easier than his stuff). Simple number theory and even properties of numbers (divisible by 3, etc.).
In terms of direct applicability to algebra, the one key proof topic is mathematical induction and the classic problems in any general algebra 2 text that are under that chapter. If you want to do something useful, covering this, even if it is pre-covering, might help them. I worry though that this is a topic that is slightly hard even for the average student within their algebra 2 class. But perhaps you could really water it down or do it gently or the like. And not make it a hill to die on but just more of giving some exposure.
Actually as I write this (sorry for stream of consciousness), I think more and more that proof ability is extremely unimportant for these kids (that probably lack computational skills). Proof classes are something that the superstars of math contest kids take in HS. Not your guys. However, you do have an alternate mission which is to awaken some interest in math. So I would let them call it a proof class but concentrate on fun activities.
One fun area is symmetry and point groups. Don't take a group theory approach to it, but instead just list the type of symmetry elements (rotations, mirror planes, etc.) and then have them catalog them on objects. If possible, have physical objects to pass around. (If you don't have enough, have the kids do teams...this is the sort of thing where teams actually makes sense, unlike computational problem solving.) You can do 3d and 2d. For 2D, you might do something with wallpaper patterns (space group) but don't do anything with 3d space groups (too hard). And by do something, I don't mean mastering it, but coloring in the birds on some Escher print or listing the rotations and planes and such that exist in some prints.
Also there was another thread on here (can't recall title) about experimental (i.e. physical) math projects for high school. Stuff like dropping the needle and finding pi. I would do some things where the kids can use their hands and such. It is summer after all. Plus really, we are physical creatures (all of us, even Andrew Wiles). And these kids are obviously not abstract thinking superstars.
Bottom line: lots of "sugar" and not much "medicine". Keep the emphasis on fun rather than accomplishment and you will end up with more accomplishment (than if you torture them with proof strategies).