We only have 1 full day on the Scottish border. What quaint village would be the best to stay for the night?
Dumfries is a small town near the Border and is near many historic sites such as Castles, Abbeys, etc. and is Robert Burns territory. I almost forgot about the Camera Obscura at the Dumfries Museum; check it out!
We did Hadrian's Wall and stayed in Hexham, England. It was great
You may want to consider Selkirk, Galashiels. or Moffat. A good place to stay would be the Gordon Arms Hotel in Yarrow. My wife and I stayed there a few years ago, and really enjoyed it. James Hogg and Sir Walter Scott met there for the last time in the 1820s.
The Gordon Arms is pretty much out in the middle of nowhere in the valley of the River Yarrow. It's a beautiful part of Scotland - not as dramatic as the Highlands, but with a serene beauty all it's own. You'll be near Traquair House, St. Mary's Loch, and the Grey Mare's Tail waterfall. Also Melrose Abbey and Abbotsford.