Short of sticking a GoPro on his head to record the games, I would recommend continue with trying to teach standard notation and avoid trying to come up with another system. Algebraic notation is the language of chess and is as fundamental as say reading and writing is to English. Just keep trying to teach it and hopefully he'll get the hang of it with practice.
Special notation sheet for young kid?

Go Pro is a great idea.
I doubt that it's legal in a tournament situation. Electronic devices are banned.

You can carefully consider the Monroi and related devices. https://en.wikipedia.org/wiki/MonRoi
Just asking. but why does he even need to notate? I mean I doubt he can analyse if he's just a 1 grader.

Your son is under 5, and it's too early for children to learn annotation. You are risking him not enjoying the game and making it work that he is forced to do.
He is not a small Adult, his brain is still developing. Wait until he is 8 or 9.

Track down his opponent after the game and take a picture of the opponent's score sheet. Maybe they can even analyze the game together with the coach help.
Or, you can do a scantron type score sheet. They general have less problems filling in bubbles. You can even automate the pgn generation: https://www.youtube.com/watch?v=nc2n-UjrzpY

Your son is under 5, and it's too early for children to learn annotation. You are risking him not enjoying the game and making it work that he is forced to do.
He is not a small Adult, his brain is still developing. Wait until he is 8 or 9.
The trouble is that once you go down that path, it may be impossible to get the kid to notate. I used to think this way, now son is 9 and doesn't want to notate, says it kills the fun... Now I think notation is a good habit to build up early, kind of like brushing teeth.
Track down his opponent after the game and take a picture of the opponent's score sheet. Maybe they can even analyze the game together with the coach help.
Or, you can do a scantron type score sheet. They general have less problems filling in bubbles. You can even automate the pgn generation: https://www.youtube.com/watch?v=nc2n-UjrzpY
I mean. If a woman randomly came up and asked if they wanted to analyse it it'd be weird. A better thing would be the kid to ask if they want to analyse
Your son is under 5, and it's too early for children to learn annotation. You are risking him not enjoying the game and making it work that he is forced to do.
He is not a small Adult, his brain is still developing. Wait until he is 8 or 9.
The trouble is that once you go down that path, it may be impossible to get the kid to notate. I used to think this way, now son is 9 and doesn't want to notate, says it kills the fun... Now I think notation is a good habit to build up early, kind of like brushing teeth.
I mean. If he's just playing chess for fun I don't see why analysing is even worth it.

The main objective is to get a copy of the score sheet.
Game analysis is secondary, but works great when both players are present; kind of like killing 2 birds with 1 stone.
Do you seriously think the kid is just going to lend her the sheet? Maybe he wants it to analyse thengame

Your son is under 5, and it's too early for children to learn annotation. You are risking him not enjoying the game and making it work that he is forced to do.
He is not a small Adult, his brain is still developing. Wait until he is 8 or 9.
The trouble is that once you go down that path, it may be impossible to get the kid to notate. I used to think this way, now son is 9 and doesn't want to notate, says it kills the fun... Now I think notation is a good habit to build up early, kind of like brushing teeth.
I mean. If he's just playing chess for fun I don't see why analysing is even worth it.
He's USCF 800 at 5, which is pretty decent. So having games analyzed by his coach can be helpful.

Do you seriously think the kid is just going to lend her the sheet? Maybe he wants it to analyse thengame
Yes, this happens at lot at scholastic tournaments in the US.
I'm writing this on behalf of my younger son (not the one whose account this is). He is in Kindergarten, has not the best fine motor, hates to notate, scribbles, unreliable, messes it up. He's about 800 USCF, and many of his opponents notate well (mostly 1st - 2nd graders). Coach is frustrated because we don't know what he doing in his games.
I had an idea of creating a custom sheet where he could circle on each move a piece type, a square letter and number, and an "x" for a capture. It would be big print. Is this silly? Any other ideas? We are at a loss.