Where to visit between Amsterdam and Hamburg?

We’ve been to both and would go back for another visit any time. For a first time visit I think I’d choose the larger, historically significant and architecturally stunning Bremen.

I wouldn't look strictly at the train stops. And it depends on your family interests. Because I am interested in contemporary art, I would go to Documenta 15 in Kassel, but that's not for everyone. Like Kassel, Cologne is out of the way for the trip, but worth two or three days, which I can't quite say about Munster, Hannover, or Bremen, each certainly good for a day and night.

I'd also point out that it's six hours straight to Hamburg, but it's also six hours (but much farther) to Berlin, an extremely rich destination. I imagine that there are excellent non-changeable, non-refundable prices 2-3 months in advance to Berlin, if you want to research them on DB. Just an idea, since you asked.

I thought the Market Square in Bremen was magical along with the Hauptbahnhof façade - then you have to bear down and wear out some shoe leather to keep the "magic" going. On a side note, Bremen is the home of Beck's beer, if that's of any interest.

Lubeck!! While not on the direct train route from Amsterdam to Hamburg, it’s a fantastic German city off the radar for most Americans. Lubeck is a historic Hanseatic League city which is one of the most architecturally-intact cities in Germany. It’s a 50-minute train ride from Hamburg and you can easily spend an entire day there or even base yourself there.
Have a great Trip!

Lübeck is a great place to visit, but it is really not between Amsterdam and Hamburg. Between Amsterdam and Hamburg I agree about Bremen is a great option.

I suggest Münster, ( or Minden, though not directly en route to Hamburg), even more so since the train stops in Osnabrück.

Why not spend the extra days in Hamburg? There is so much to see and do in HH, or you can base there and do day trips to towns like Glückstadt or Lüneburg or even St. Peter-Ording. Or go apple picking in Jork (truly underrated). Take a day trip to Helgoland or Bremen or Lübeck or Schwerin.

From a travel perspective it is much simpler to base in Hamburg than to keep transferring, finding luggage lockers, carrying stuff, etc. and there is no way the RS tour will do Hamburg itself justice, let alone its surroundings. But if you are looking at breaking up the train ride for whatever reason, Bremen would be my recommendation.

You haven't mentioned what your interests are, which might help us help you narrow it down.