When I cycle, my lower back gets sore after about an hour. It recovers if I get off the bike and stretch backwards.

Are there any exercises to strengthen my lower back?

deadlifts, back extensions, good mornings, RDLs, etc.

Are you a competitive cyclist? If not, it might be time to switch up your cardio. Alot of people get this problem due to the spinal flexion when you cycle.

I had a lot of lower back pain after I injured my knee many years ago, mostly just a painful stiffness. I found a combination of stretching and heavy SLDL's helped me a lot. Stretching exercises I would recommend: toe touches, ballet dancer stretches, stretch kicks, splits, tai chi swinging arms. A lot of the stretching I learned studying Karate and Tai Chi. You might want to try out some martial art schools if you have time for it.