Since the requirement is "short and cheap" I'd second the recommendations for either Antwerp or Cologne.

In both cases the trip is relatively short and others have already found pricing that fits your needs.

In Antwerp, the train station is easy walking distance for the center of the city, which has a handful of quite nice museums, an impressive Cathedral, and decent shopping, including designer shops as Antwerp is somewhat of a design center. Oh, there ARE also diamonds!

The train station in Cologne is also very central - you could hit the Cathedral (Dom) with a rock from the train platform, and there is a very nice Roman museum right there also. For dinner, Fruh which is right there, is fun.

I didn't check train schedules, but would think that in either case you could catch a morning train and then one back to Amsterdam after dinner.

If you want to stay in the Netherlands, Den Haag is a quick train ride and is very nice as well.