PlayStation 2 , when disassembled, consists of series of Flex Cables (Ribbon Cable). In case of MemoryCard slots, one ribbon cable series is used. If, your one slot is not working as it should while the other is functioning normally, do the following:
1) Take a vacuum cleaner, attach its small hose, and suck it through that slot which is not working.
2) Disassemble the upper cover of PS2 by removing all the screws from the base of the covering. When the covering is removed, remove the two small philip head screws on the slots. You will see a ribbon cable sequence there. Remove it, take an eraser, clean its contact points and again place it back.
Note: First, perform the Step 1. If it doesn't help, then proceed to the second step. Otherwise, it is risky that PS2 flex cable can be damaged.