hypnodean--
Depends on your idea of fun.
For food and people watching, downtown has CityPlace, a grouping of restaurants, shops, movie theater, bars, Comedy Club etc. There's a trolley that will take you (for free) between there and the rest of downtown--for drinks on the water (intracoastal, not ocean), Bradley's is usually the place.
For more sedate fun, the Flagler Museum is just over the bridge, only open on weekdays; or just take a drive up and down the "island"...see Trump's house, Estee Lauder, my personal favorite Jimmy Buffett, etc., and gander at all the rich people's hangouts.
A little more of a drive away, there are several orange groves just a little south of West Palm (Boynton/Lake Worth) that I often take out of town visitors to...they give tram tours through the groves and have the fresh fruit stands, if that's your thing.
There's also tours of the Everglades, but the furthest north option closest to West Palm is still about an hour away....
But if you're just looking for "hanging around" options, I'd definitely recommend CityPlace...a bit overpriced, but ok for the short term.