Ok, remember that the GP-Wiz49 reports your 49-way joystick to the computer as an analog joystick.
MAME's deadzone adjustment was designed for analog PC joysticks. It tells the software to ignore any analog stick movement, in any direction, for a limited distance from exact center. The farther you move the slider, the further from center you can move your analog stick without making your character move in the game. That way, if you're using a really sensitive PC stick, and your arm twitches a little tiny bit off of dead center, you don't move accidentally.
Your GP49 has multiple modes: Raw49, Scaled49, 8-way, 4-way, 4-way Diag, 2-way Vert, 2-way Horiz, and 16-way. In any of the 8, 4, 2, and 16-way modes, the GP49 will always report to the PC as an analog joystick pushed all the way to the edge of it's travel. Therefore, in any of those modes, the deadzone setting will not do a darn thing, since the PC only ignores joystick presses that are near the center point.
However, in raw49 and Scaled49, the GP-49 takes advantage of your 49-way joystick's ability to report how far from center it's been moved, and will report to the PC as an analog stick that's only moved partway. That allows MAME's deadzone setting to come into play. If the stick is barely moved away from center (like yours are when they fail to center up all the way), then MAME will ignore it- and your son doesn't crash and die. The downside to this is that you lose the fancy 2, 4, 8 & 16 way modes, and 4-way games especially will suffer from not being played in the 4-way mode. 8-way games may not be quite as good as they are in 8-way mode, but still ought to be pretty playable. Doing this is not meant to be a permanent solution- it's just until your grommets get "broken in", or you try out the stronger grommets I mentioned, or you figure out some other solution to get your sticks centering better.
Hope that helps!
-Kremmit