When was the last time you checked the health of your battery? The battery in the Jeep, not the remote. I can say from experience that with today's cars being very dependent on computer systems, low or erratic voltages from the battery / charging system can cause all kinds of weird stuff to happen. On my wife's Nissan, the speedometer all of a sudden started reading 3X the real value; was kind of fun going down the street with the speedo pegged on 140 mph! One of my neighbors had the right side sliding door on his Honda Odyssey start opening and closing randomly. In both instances, replacing the battery with a fresh one fixed the issue. If your battery is more than about 4 years old or so, this could be the source of your problem.
Keith