Is there a way to strengthen individual pieces?
Those joints are based solely on friction, which comes from:


the "bite" of the socket - how rigid the containing socket is
the rigidity = non-elasticity of the ball
the shape of the ball 
the relative friction between the two surfaces


The old plastic gets weaker with time, which rules out fixing the first two causes.

The shape of the ball would be pretty hard to fix; also the shape is not likely to have changed much.

This leaves the friction itself. So what you could do:


spray an adhesive on the ball to make it tackier
glue the joint in place (KRAGLE!!)