What makes kaons strange apart from the fact that they have strange particles in them?
Noting. A hadron, including strange quarks or antiquarks is called to carry "strangeness". It makes sense to classify particles according to their flavor, because flavor is conserved under the strong interaction (but of course not under weak interactions). "Strangeness" is a "charge-like" quantum number that is conserved as long as you neglect weak interactions.
They were called strange (and got a "Strangeness number") at a time where the quark model did not exist yet, because they had an unexpected long lifetime. This got explained with the quark model and the quark was named after the particle property. To add some strangeness, the strange quark has a strangeness of minus one.
Yeah, the negatively charged quarks all have also negative flavor ;-).
The Wikipedia article on kaons tells a bit about the history of the concept of "strangeness":
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