The usual problem with stirring hotplates is that the top is larger than the container that you want to use, so there is a fairly small area of contact between the container and the plate (which is usually three points of actual contact and a thin air gap everywhere else). The container lose heat from the sides and top, so the plate has to be pretty hot to keep the container temperature up.
One solution to this is to put the small container in a water bath (such as an aluminum pan) that is about the size of the hotplate. To minimize evaporation, make an aluminum-foil cover for the water, or substitute sand for water. The idea is to have a large mass that won't change temperature quickly, and let the small container "go along for the ride". If you want to use a magnetic stirrer, the distance from the hotplate top and the magnet can't be too large, but an aluminum cake pan and glass beaker should be well within range.
If you didn't need the stirrer, a thrift-store crockpot with an incandescent light dimmer would make a great water bath. Manually holding 110-120F should be no problem at all, as long as your room temperature and air circulation don't vary too much.
Lessee....for a really cheap stirrer, maybe a thrift-store electric mixer, or a line-powered variable-speed electric drill? Cut away most of the blades on a mixer beater, bend the remaining tabs into propeller blades, and you could be all set in no time.