When I was a child, the first thing that made sense to me was that "x is the number that makes the statement true". That is, 100 = 100, so if 25 * x = 100, you need to figure out what value of x multiplied by 25 gives you 100. I didn't even understand the manipulations or how you would really solve for x at first, but I did understand that you were trying to figure out the number that would do that. I would just phrase it like a puzzle, like "what plus 9 equals 14" or "3 times what equals 42"?

For some reason, one of my earliest memories I have is from when I would try to show off to older kids that I understood algebra.

Have you tried analogous approaches, such as showing kids an empty box, a question mark, multiple metaphors, etc?

I find that visual approaches work well, but when you write an English letter which has already been used for other purposes, it's a little less clear.

I've always translated "x" to "?" (the value to be discovered) as a mental shortcut. The "?" symbol might be easier to interpret for third graders.

I think they are used to seeing empty boxes for mystery values.

Have you looked at the Dragon Box app?

I shill DragonBox every chance I get ;) It does a great job of teaching abstract algebra to children who don’t even know arithmetic.

I think it helped my son for me to explain that letters were just short "names" in place of the phrase "mystery number." Or something along those lines.

Like "We are just going to call it X or Y instead of the unknown number we are trying to figure out."