I don't have a specific system, but one thing I highly recommend trying is the Montessori method of teaching math concepts. Doesn't have to be done using Montessori concepts generally, just the system itself is really effective for kids who don't "get it" when explained verbally or on paper.
The method is using physical objects to represent the mathematical concepts. It typically is used starting at 3 years old, and it's normal for kids in Montessori to be doing multiple place value multiplication before 6, due to the simplicity with which this works.
There's quite a few explanations of this method out there, for example. Most likely he'd want to start around the Bead Chain I suspect, as that's where the beginning concepts around place value are taught (mostly later, but this is an important first step). You don't have to get the actual materials - they can be expensive - but you can replicate this yourself.
Basically, you have a chain of beads for each number - 1's, 2's, 3's, etc. - that goes up to that number squared, separated into groups of that number. 7's has 49 beads in groups of 7. Obviously 1 is very short :) This gets the child used to counting 'by 7' etc.; they can literally just count along it, and make a marker for every multiple of each number - and place that marker at the 7th for example. So the 7-chain would be laid out, they'd count 1,2,3,4,5,6,7, put a 7 marker, then 8,9,10,11,12,13,14, place a 14 marker, etc. up to 49. You can see this is teaching multiplication concepts already, but in a way that makes sense to a preschooler (but just as useful for an elementary aged child who's not there yet).
Then there's also the "square" bead chains for each, which have the squares of each number in actual squares. 7 square has 7 49-squares. The 10 square has 10 100-squares.
The next major step is the stamp game, which is where place value is really solidly taught. You can use the Bead Tray intermediary with the 10s (10 cube = 1000, 10 square = 100, 10 chain = 10, 1 bead = 1) to help with this. They start out just making a number out of "stamps" (basically, printed out digits), so 4785 is 4 1000 stamps, 7 100 stamps, 8 10 stamps, and 5 1 stamps. Then, you move on to using the stamp game to do basic addition - so 4324+2513 - by making the two smaller numbers from their constituent stamps, then adding them by counting what's left - and then you can teach carrying/etc. also.
The next step that's really nice to move on to is the bead frame. The small bead frame is great for addition, you have a larger one for multiplication. This is sort of like an abacus, and is used to teach the decimal place.
All of these can be made at home if you want, the materials are easy to find but not super cheap. Do try to follow the color coding they do - or use your own, but make it match; it's really easy for the kids to see the patterns between the different things when the colors match.
There's further materials available for teaching multiplication and division; the long division materials in particular (racks and tubes) is really neat in that it will look like it makes zero sense to an adult who's not familiar with it, and even when it's explained it will not make any sense how it's connected to long division, but it both works and kids immediately understand long division once they learn it!
One recommendation I have on materials, which I used in the 2020 spring to help my Montessori 2nd grader - something as simple as colored beads (the kind you'd get to fill a vase with a fake plant) will do for the stamp game, and you could do a similar activity to the bead frame [by only giving him 10 of each color] with them. If I were to spend money, I'd spend it first on the bead chains - those are harder to make, and they're really important for laying the foundation of multiplication - and see if just colored glass beads (which may already be around the house!) would work.
Legos or similar also work for some of these - might even be a replacement for the bead chain if you want to go super cheap. Each lego is whatever number of dots it has on top - so 4x2 legos are 8's, etc. Most numbers below 10 have a lego that can work for it. 1,2,3,4,6,8 are really easy, and I think only 7 is truly difficult to find.
