You should be able to get a Thin Terminal device for about $100US with USB3.0 and gigabit ethernet and/or wifi. With a little shopping, you may even be able to get one with a low profile PCIe slot for similar money. A few examples of such machines are the HP T5740, t520 or t620. Dell/Wyse also should have acceptable options.
Choosing a Thin terminal should cost far less than a NUC while being much more capable than a raspberry pi for not much more money (a raspberry pi 3 still needs a uSD card, a case, and an unusually powerful USB PSU).