Can not get new Apple Watch 3 to pair with my wife’s iPhone 6

Apple Holiday Return Policy (Purchase between: 15-Nov-2019 thru 25-Dec-2019, Return by 8-Jan-2020)

All purchases made before November 15, 2019 and after December 25, 2019, are subject to the Standard 14-day Return Policy. 

https://www.apple.com/shop/help/returns_refund

If it's a new apple watch 3 then it will have watchos6,this requires ios13 which means an iphone 6s or higher.

You can't change the watchos to a lower one so yes your best option would be a newer phone if you can afford it or you could pick up an iphone 6s fairly cheaply.