Under any plastic D-Pad post-NES-era (which includes original GameBoy) there is usually a rubber disc/cross. Yours may have rotted out and "flattened" in it's spring-cup areas for whatever reason.
I'm not sure if there are any companies out there that sell replacements (though there very well may be) ...
you could actively seek a busted GB with a still-firm-rubber part and gut it for a working replacement. (Of course you'll also need an NES-bit screwdriver.)