To amend your tax return, use Form 1040X, available on the IRS website. You can correct your Social Security number at the top of the form next to your name or the number for a dependent on line 29. Explain why you're filing the amended return in Part III of Form 1040X. Your explanation doesn't have to be complicated; just saying "Filing to correct the Social Security number for my dependent child Susan" is sufficient.
Filing Form 1040X
You must fill out a separate Form 1040X for each tax year you need to correct. On the bright side, since you're only correcting information, you don't need to fill out all the line items. Just enter your correct information at the start of the form and explain what you're changing in Part III.