You would have a household employee. You are not required to pay household employee tax or withhold social security when the employee is your child under age 21. You are not required to withhold federal income tax. As long as you don't withhold federal income tax and you don't pay household employer tax, you are not required to issue a W-2.
https://www.irs.gov/taxtopics/tc756
If you want to issue a W-2 so the "wages" are documented, you must first get an EIN, employer ID number. You don't use your SSN as the payor if you pay wages.
If you don't issue a W-2, your son can file a tax return by including the wages and noting "HSH" on line 1 next to the dollar amount. Turbotax has a place under "other uncommon income" to report household employee wages. If the wages are less than $4300, he would not be required to file a return at all.
I think a more important question is whether the IRS would view this as a sham to turn an allowance into "compensation" for purposes of the IRA contribution. I would feel a lot happier about your son earning $25 for cutting the neighbor's grass and reporting as schedule C self-employment income, rather than being paid by his parents. You may be taking a risk here, and while $300 per year is probably not worth paying for a professional opinion which might cost $300, the situation seems like a vulnerability to you.
(Then I would ask, if these really are "wages", are you subject to your state laws regarding minimum wage, child labor, hazardous occupations, and so on. Are you creating another legal vulnerability.)
