An ideal exercise for fostering good sleep is one that taxes both your body and brain. There is another criteria that I will add: The exercise should be reasonably safe. The goal of this website, after all, is to help you sleep like the dead, not be dead.
Bicycling either on road or off road (mountain biking) is good exercise, especially for your lower body. It usually puts less stress on your knees, ankles, hips and back than jogging does.
It can also work your mind in a few ways. The changing scenery of bike riding keeps your mind stimulated. More importantly, bike riding, especially mountain biking, forces your mind to pay close attention to what you are doing or risk serious bodily injury.
And that brings me to the issue of safety. Nearly 600,000 people go the emergency room annually to treat bicycling-related injuries. That's a big number. And think of all the people who were injured on a bike and did not go to the emergency room.
What this means is that there is a pretty good chance that, even if you just ride lightly around the neighborhood, you will crash sooner or later for one reason or another. Trying to get a good night's sleep with bruises, scraps or worse is not easy. Trust me, I know.
I'm not saying to avoid riding a bike. If you enjoy bicycling then do it. Just understand that there are arguably safer and more effective ways to exercise enough so that you can sleep well.
External resources
Kenkifer.com article
Bupa article
Choose Fitness article