I am re-creating an old ADC module, which is connected by 1.04mm (0.04") pins to the main board, they are around 5mm long from the top surface and they go into metal inserts which are soldered to the main board.
Any idea where I could get these pins or how are they called?

 

Mouser has lots of them in many styles, pin sizes etc.

EDIT: whoops, thought you meant the sockets. Try these.