Land Vehicles
The most common use for an old weed trimmer motor is to power a bicycle. Some people have used them for powering skateboards, but this has been proved to be dangerous and is definitely not recommended. Most weed trimmer motors are a little small to power go carts, unless the vehicle is for a small child. To power a bike with an old weed trimmer motor, remove the attachment to the rotating shaft and replace it with a spur gear. Run the bike chain around the spur gear and to the gear on the back axle of the bicycle.
Marine Vehicles
If the shaft of the weed trimmer is in good shape, you can make a small outboard motor by replacing the weed whacking mechanism with a propeller. This makes an excellent outboard motor for a small craft like a single person dingy. It is also useful for maneuvering a sailboat into dock--because the outboard motor is not attached to the boat it can be reversed by rotating the shaft. It is a handy auxiliary motor to have if you are stuck on a lake with no wind and need to get into port, or on a fishing boat when your main motor conks out.
Model Aircraft
A small weed trimmer motor makes an excellent power source for a radio controlled model airplane or helicopter. The engine block is usually aluminum because it is important to keep the weight down to prevent fatigue when whacking weeds. This design means that the weight of a weed trimmer motor is as low as the motors designed for radio controlled models, and it is a lot cheaper.
Other Uses
One interesting use of old weed trimmer motors is making a homemade cordless blender. For some reason, electric blender motors seem to burn out before the rest of the blender wears out and weed trimmer motors last longer than the rest of the weed trimmer--two problems that solve each other. This is a useful device if you go camping or to tailgate parties and someone in your group likes blended drinks. Another use is making a battery charger by connecting the motor to an old car alternator.