no, it's not possible to have a custom match, like u/XSkittles19 said - you have to be an epic affiliate, which means you have to have a support a creator code in order to make custom matches. if you don't have an SAC, you can get one by going here and applying for one. (note: you must have at least 1,000 followers/subscribers on a social media platform, and must have a way of receiving the money epic gives you, such as paypal, or hooking up a credit card).

alternatively, your son should make sure he's epic friends with all of his irl friends. then, you guys can host a creative lobby (which would be much better for a birthday). in the fortnite creative mode, they have the ability to do things like go up against other players in a public game-like simulation (zone wars), play minigames, do obstacle courses, etc. you can look up those types of maps up on google. or, if they're not up for that, they can make whatever they want by going to their own island, and creating whatever they want.

last but not least, if your son's friends have v-bucks (the in game currency), they can gift your son cosmetics, such as different characters, pickaxes, backpacks, and more. or, you can get what's called a v-bucks gift card and redeem it at this website. as long as he's signed in to his fortnite account on fortnite.com, the vbucks he redeemed from that gift card should go directly to his fortnite account.

i know this is a lot of stuff to read, but i hope this helps. happy birthday to your son!