1) The game will determine what size volume pot you need. Some use 300-om and some use 50k-ohm. A 5k-ohm pot will work for most games.
2) Cut one of the wires to the speaker.
3) Connect one side of the cut to the center tab of the pot.
4) Connect the other side of the cut to one of outer tabs. If you want it to turn right to go up and left to go down, use your meter to find out which side measures 5k-ohms between that tab and the center when the knob is all the way to the left and connect it to that tab.
5) You can now turn the volume up and down. If it won't go completely silent, then you need a a larger value pot. If the volume goes from silent to loud with only a little rotation, then you need to go with a lower value pot.
Bear in mind that some pcbs will have a volume pot on the board, some will have an external volume that connects as above, and some will have an on-screen volume adjustment...