My wife received a 1099-NEC for her expenses on my business trip. Not one of the 6 types of income listed on the free online Turbotax for 1099-NECs. Where do I put it?

You will enter it in the section for the 1099-NEC and then you can choose this is not money earned as an employee or self-employed, it is from a sporadic activity or hobby.  The answer to this question is not sent to the IRS, it is just used for the program to determine where to enter the income on your return.  

The 1099-NEC is not a form that will allow you to stay in the free version regardless of how you enter it.  You would need to use Deluxe or higher for the 1099-NEC.