Low RPM, Low Power, Low Torque Motor (to be solar powered)
If you can do some mechanical calculations for your system, determine the power needed to drive your load. This is the most important part of selecting the right motor.

It is very unlikely that you will be able to find a small motor that spins with such a slow native speed. Do you want to vary the drive speed from 5 to 30 rpm or will it be fixed somewhere in that range? 
My recommendation would be to pick a small motor that has enough power for your application and then gear it down to speed. If you want speed control a simple off the shelf pwm driver should be fine, set the speed and forget it. For more advanced control an arduino project might be in order.