If you'd just like the final numbers and are less concerned about the reasoning behind them, answer is as follows:
EITHER get 2 African clawed frogs
OR 6 neon tetras, 6 cherry barbs, 6 gold barbs, 3 guppies, and 0-1 snails.
Detailed answer below:
African clawed frogs are generally recommended to be kept in a species-only tank. (As in no fish, just frogs.)
Just so you know, African clawed frogs will eat guppies and probably most of the other fish as well. The guppies will likely breed if given half a chance, so this could be desirable, but I would count on the tetras and barbs getting wiped out as soon as the frog is big enough to swallow them. If your son (or you) would be upset by seeing his fish being eaten by his frog, then I would strongly recommend getting either frogs OR fish and not both.
Also select your substrate carefully. There's a good chance that the frog will ingest some of it at some point and if the substrate is too large or sharp it could do severe (potentially fatal) damage to the frog's digestive tract. Either select a substrate that will be too large for the adult frog to fit in its mouth or small enough and smooth enough that it will have a chance of passing through the digestive tract without doing damage.
Pool filter sand would likely be safe for the frog but could potentially clog your filter. (I use pool filter sand in my tank and haven't had any issues, but I also don't have a frog kicking it up into the water column so your mileage may vary.)
Apple/mystery snails can get fairly large and have a pretty high bio-load, so just be aware of that. They may end up putting nitrates into the water (which will feed algal growth) faster than they can eat the algae when they get large.
Also, like the substrate, they could pose a hazard to the frog if they get swallowed at an awkward size.
And, if you get more than one apple snail, you may find yourself with a snail population explosion. Again, this could be desirable, but then again, it may not.
Tetras generally do best in groups of at least six. (Speaking from experience, I can say that I say a dramatic change in behaviour when I went from four black neons to eight.)
Guppies are generally best kept in ratios of at least two females to one male. (Unless you don't want them breeding in which case you might want to consider all females.)
Barbs should generally be kept in groups of at least six. They may fin-nip the guppies, especially if you end up with a variety with particularly long fins (veiltails). Keeping them in larger groups (12+) generally reduces fin-nipping. However, I wouldn't recommend putting that many fish in a tank this size.
Even with the minimum recommended numbers of each species (6 of each type of barb, 6 tetras, 3 guppies, 1 frog, 1 snail) and a good filter, you'd likely be looking at doing 50% water changes twice a week.
If your son is more interested in the frogs, then you could quite happily house a pair of African clawed frogs in a tank that size.
If he is more interested in fish, then you could put all four of those fish species together in that tank (6 cherry barbs, 6 gold barbs, 6 neon tetras, 3 guppies). However, I'd still recommend that you keep a pretty close eye on water parameters (weekly testing) and consider doing at least a weekly 50% water change.
If there's one particular fish that your son would like more of, then maybe you could discuss going with only three species instead of four to allow for more of whichever one he's most interested in. (All of the fish you mentioned would also do well in larger groups.)
If you do decide to get both male and female guppies, try to come up with a plan for what you will do with the surviving fry. (Some may get eaten or sucked into the filter, but some will more than likely survive.)