You don't need any proof at the time you file your tax return. That comes later if you're audited.
You report the income from your own records, no matter how weak those records are. The IRS considers cash income (no W-2 or 1099-MISC), for work performed, to be self employment income. Enter at "Business Income & Expenses" and TurboTax (TT) will complete Schedule C or C-EZ for you and allow you to deduct any expenses associated with this income. You'll also have to pay self employment tax (social security & medicare tax) on any profits greater than $400.
