Petroleum based lubricants you need to be careful with, since they can act as solvents. Zhaph mentioned non-penetrating machine oil, this sounds like a safe bet. If you are in doubt, grab a gnawed brick and submerge it in a sample for a few days and see what happens before putting it on your precious train parts. Science!
Second hand info:
I have heard that ...
Generally for this sort of thing the recommendation is a "light machine oil" (the wording from the Hornby Maintenance sheets). These are "non-penetrating" and so won't damage the plastics which might happen with 3-in-1 or WD-40 style oils.
This can often be found as "Sewing Machine Oil".
Fleischmann (a model railway brand) sell ...
I suggest you contact LEGO support. The set is recent enough for them to have spare parts. They'll happily replace the Push & Go component if you can demonstrate the part is defective and you own the set. To prove you own the set you'll need a code in the instruction booklet.