When I came across these tiny articulated 'desk lamp' reading lights, the white version instantly reminded me of Pixar's Luxo jr. .
The lamp can be switched on and off by a tiny push button on the back of the lamp and its arm has two joints for adjusting it vertically. Unfortunately it can not swivel sideways, making it unable to do that famous Luxo jr. pose, so I decided to modify it.
I added a swivel joint by making a cut between the lamp and the upper part of the arm, then drilling a hole through the center of both ends and inserting a plastic pin to join the lamp to the arm.
First step is to take the upper joint apart to remove the lamp from its arm. Just pull both 'thumbwheels' apart, one has a small pin that locks into the other, the other has a hollow tube that acts as a hinge pin.
Then take the lamp apart by twisting the bulbous hood counter-clockwise (with the lamp pointing to the left, rotate the top away from you), then pull it gently off of the rear section. Carefully remove the black lamp interior which holds the LED, batteries and switch from the rear section.
Make a pin by heating the end of a piece of thin plastic rod over a heat source, once the plastic rod starts to melt push the rod against a smooth flat surface to form the flat head of the pin.
I made the pin from a thin piece of frame of a set of Yellow Submarine joints, I don't know what type of plastic it is made of but it is very resilient, much more so than polystyrene.
Measure the diameter of the pin to determine the size of the holes to be drilled.
Now use a razor saw to make a cut between the lamp and the hinge (blue line marked '1' on detail picture).
Make sure both ends of the cut are smooth, then drill a hole for the pin in the center of both ends, make the hole in the rear section of the lamp slightly oversized to allow the pin to rotate smoothly.
Make the hole in the lower part with the hinge to fit the pin as tight as possible.
Insert the pin in the rear section (position of pin is marked in violet on detail picture) and check if the head of the pin does not obstruct fitting the black lamp interior (if so remove the pin, sand down its head a bit and try again). Then push the lower part on the pin (if the pin does not fit tight enough to hold the joint together, remove the lower part, put a small drop of CA glue on the end of the pin and refit).
Make sure the joint swivels properly.
I also added a thin slice of soft vinyl tubing (marked '2' on detail picture) to the upper joint to keep it from getting loose (do note that adding this slice makes assembling the joint much more fiddly !).
Finally put the lamp and the joint back together again.