Kelso (20km northeast) — Kelso Abbey and Floors Castle, the largest inhabited castle in Scotland. An easy half-day from Jedburgh.
Melrose (20km northwest) — Melrose Abbey, one of the finest of the four Border Abbeys, and nearby Abbotsford, Walter Scott's home. The two together make a full day.
Dryburgh Abbey (15km northwest) — The most atmospheric of the Border Abbeys, set in a loop of the River Tweed. Sir Walter Scott is buried here. Compact and worth an hour.
Edinburgh (75km, ~1.5h) — A full day's excursion from Jedburgh, or the natural start or end point of a Borders trip.