What is the source of electrons that flow through a wire with a battery and light bulb connected in a simple circuit?

The electrons come from the metal of the wire, light bulb filament, and plates inside the battery. In other Words, they are already everywhere in the circuit, even before it is connected together.

The electrons are made to move (and thus do work, e.g. heating the filament to make light) by the electromotive force generated by the corrosion of the metal plates inside the battery by the electrolyte between those plates.

Note: while the electric current flows around the circuit at a speed close to the speed of light, individual electrons within that current actually move very slowly due to the fact they keep bouncing off atoms and other electrons thus performing a "random walk". These speeds are on the order of millimeters per hour.