I hope you don't mind my saying this, but I'd get him to go alone (unless you want to go for the holiday of course). He doesn't need help at this age unless he actually gets into a pickle & needs to start doing this stuff independently, it's all part of the fun to muddle through by yourself and it's how you gain confidence. Think it's a brilliant idea to take a gap year, he'll have a real advantage when he starts uni, and Paris is an amazing city to work in.
Internships ('stages') in France nearly always require a Convention de Stage to be signed by a HE institution, as the stage has to be within the frame of some kind of studies, but CDDs (contrat à durée déterminée - fixed term contract) don't. All stages which last beyond two months must be renumerated (starting from the first day) - the actual pay can vary vastly but you usually also get a discounted Navigo and around 35-40€ worth of meal vouchers/week. CDDs offer similar, but I think pay is min around 1500€ ('le SMIC' for 2018) which translates into around 1200€ net. If he does find a stage he likes the look of, he could see if his future uni will it, explain the situation to employer and see if they'll take the risk of hiring an intern who isn't technically a student yet (my old uni signed for a stage I completed after I graduated and employer was on board - authorities didn't check up - my cousin & her uni worked out something similar). Jobmania.fr good for student jobs. Could look at expat or travel insurance companies such as Europ Assistance.
I think he could probably support himself doing babysitting through Speaking Agency as they pay well and have a lot of clients on their books (I had a great experience with them), but I suppose if he doesn't want to au pair, that would probably only be a stopgap.
Oh and good idea to put a photo on his CV. I know it's not done here, but it is in France (have been involved in recruitment and everyone includes a photo). And worth getting someone to thoroughly check over his CV and cover letter - if the employer is not specifically recruiting a native English/non-French speaker, they probably won't be lenient re: Anglicisms or mistakes. When emailing, 'faut mettre les formes' - this has some good examples of lettre de motivation-speak.
Oh and French like it when someone describes themselves in these letters as 'sérieux' - it's a false friend and means reliable, conscientious etc etc (over here I see ads where someone's got perfect English but has described themselves as a 'serious girl', dead giveaway that they're French!).
I know people who got Air BnBs, but you could also check out 'chambres d'étudiants' in the areas near universities or schools offering prépas (loads of older people with massive apartments renting out rooms in the 16th near Saint-Jean for example) - usually around 500€ with bills, and you can live in lovely homes in fab neighbourhoods you couldn't normally afford (otoh, being a lodger in someone else's home isn't all plain sailing). I also had luck with Erasmusu (in another Euro country, but just to reassure that the site is above board).
Hope he finds something great!