You can get decent pre-finished hardwood for around $4 a square foot plus installation, and if this is upstairs or on a crawl space with a wooden subfloor, installation could be DIY pretty easily. Naildown isn't that hard. That's not too expensive in the flooring world, and there aren't that many choices that are less expensive that would be asthma and budget friendly.

If you are downstairs on a slab foundation, you could stain it and finish it. You will need some products with VOC's though, and your son may not deal with that so well depending on the severity of his issues. There's vinyl, either luxury vinyl tile or some type of sheet vinyl. Those can also emit VOC's when new, so you'd need to research any low VOC options as well as assess their effect on your son also. There's cork, but it's more expensive than hardwood. There's tile, and you can even get tile that looks like hardwood if you wanted, but that's going to be the price of wood also, and leave you using some rugs in the winter because it's cold. Unless you install underfloor heating cables, which is WONDERFUL, but now you're well above the price of wood again, even DIYing the install.

Really, wood would probably end up being the cheapest and safest choice, especially if you don't have any DIY ability.

And, you don't want tile because.......?

We have wood-look vinyl plank. It looks better than all but the most expensive laminate and doesn't come with the installation issues such as need for underlayment, snap and crackle noises etc. It is very easy to clean and allergy friendly.

We got ours on clearance for 80 cents/sq ft from Lowes but this one is now discontinued, similar products are around $1-$2 sq ft and it is an easy DIY install. All you need to install it is a sharp carpet knife and measuring tape. I've never done flooring before and I did 1500 sq ft in 4 days all by myself, and it turned out perfect, I watched a youtube video to learn how.

Here's ours - we'd just moved in so things were not really arranged yet but this shows the floor OK.

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Tile will be the best option for health. With an underfloor warming system. Or washable rugs.

Have you thought about cork. That is what I am putting in our bedroom. I hate carpet, it can never be clean enough if you ask me. And the rest of the house is hardwood.

Cork runs from inexpensive to expensive depending on which you choose. Once you seal it it is waterproof. And it is warm to the touch, unlike tile.

Marmoleum is a wonderful flooring material. We have sheet in our bathroom, and checkerboard click tiles in our kitchen. It feels great under foot and the click version is insulated with cork backing- not too cold in winter. You can do designs in it if you want a fun look.

I second the cork idea. There are loads of options. If you go onto houzz.com, you can see some examples of rooms with it and also examples of the different types.