I'd be careful with lacquer thinner soaking as it may dissolve or permanently damage the rubber pad.
A quick Google search gives a number of solutions for softening old hardened rubber including soaking the object in a 3:1 solution of 91% rubbing alcohol and wintergreen oil, or applying liquid silicone to restore the lost plasticizers. Short applications of Glycerin was another solution, but there were warnings that prolonged soaks in glycerin would turn rubber to mush. Another one suggests using a mixture of wintergreen oil and toluene. But toluene is a pretty potent organic solvent too, one of the components of PVC pipe cleaner. Some folks were keeping their go-kart tires soft for better traction by spraying them with WD-40 and wrapping them with saran wrap to keep the WD-40 from quickly flashing off.
