Remove the left over deodorant bits from the stick. Pull out the screw mechanism which drives the deodorant up to your armpit.
Now you've got an empty chassis into which you can insert the battery and circuit board from your solar light.
Disassemble the solar light. It should just be a solar cell wired to a circuit board, and a battery pack.
Keep an eye out for the light sensor. A little cadmium cell that is usually somewhere near the solar panel. The cadmium cell acts as a switch, so the light turns on in darkness and stays off in light while the battery charges. I chose not to wire an on/off switch since the battery usually stays charged until at least midnight after a day in the sun and I wanted to keep this light simple.
You could wire in a switch (and use the spinny bit on the deodorant stick for a weatherproof switch actuator if you like late night (early morning) bike rides.