In 2015 my then 76 year old wife earned $53,000 in additional income (fee for aacting as executor of her aunt's estate ). Does she have to pay social security tax on that money at her age ?

No age is a barrier to paying social security tax on earned income.  However, the fee earned by a non-professional for handling a family estate is not subject to self-employment (social security) tax.  

You can report it as other income on Form 1040 line 21. Here's how to enter it in TurboTax.

• Click the Federal Taxes tab ( Personal in the Home & Business version)
• Click Wages & Income. (Personal income in the H & B version)
• Click "I'll choose what I work on."
• On the screen "Your 2015 Income Summary," scroll all the way down to the last section, "Less Common Income."
• Click the Start or Update button for the last topic, "Miscellaneous Income, 1099-A, 1099-C."
• On the next screen, "Let's Work on Any Miscellaneous Income," scroll down and click the Start or Update button for the last topic, "Other reportable income."
• The next screen asks, "Did you receive any other taxable income." Click Yes.
• On the next screen, "Other Taxable Income," enter a description and the amount. Click Continue.
• On the next screen click Done.

The income will be reported on Form 1040 line 21 with the description that you entered.

Given the amount, she may end up owing an underpayment penalty.