I received a 1099-G form, stating the amount of my overpayment during 2016. When I try to enter this form in TurboTax, the overpayment is not an option to select when it asks the type of payment reported.
You are entering the 1099-G in the wrong place. Do not enter it under "Other 1099-G Income." There is a specific topic for state income tax refunds. Here's how to enter a state income tax refund in TurboTax.
Click the Federal Taxes tab.
Click Wages & Income.
Click "I'll choose what I work on" or "Jump to a full list."
On the screen "Your 2016 Income Summary," scroll down to the section "1099-MISC and Other Common Income."
Click the Start, Update, or Revisit button for "State and Local Tax Refunds on Form 1099-G."