What is the property, whose violation led to the assumption of strangeness? Prior to the discovery of strangeness was it assumed that particles that are produced by strong interactions can decay only through strong interactions? But the discovery of particles like kaons, and baryons showed that they were generated by strong interaction but decayed through weak interaction. Is this the property that is called strangeness?
There are six types (flavours) of quarks in the world. u- and d-quarks are the lightest and protons, neutrons and light mesons (i.e. the hadrons that are long-living) are composed of those two. The next quark - s-quark from "strange quark", is much heavier (95 MeV compared to ~2 MeV and ~5 MeV for u and d respectively)
The electromagnetic and strong interaction conserve the flavour. That means that e.g. if the s-quark radiates or absorbs the photon or gluon it remains to be the s-quark. Thus if there were no weak interaction the s-quark (or more correctly, the lightest hadron that has it as a constituent) would be stable and never decayed simply because there would be no process to provide such decay.
You can introduce the conservation law of the number of the s-quarks. It is this conserved number that is called "strangeness". If you collide the light particles with enough energy you may produce the s-quark - anti-s-quark pair. Thus you could produce the pairs (to conserve total zero strangeness) of these heavy hadrons with s-quark that would however never decay by their own.
In real world there is weak interaction which changes the flavour i.e. s-quark may radiate W-boson and turn into u-quark. The weak interaction allows to all hadrons (except proton) to decay through new processes and for your question "was it assumed that particles that are produced by strong interactions can decay only through strong interactions?" the answer is firm NO - the weak interaction was discovered exactly in the hadron decays. More important for us now is that the weak interaction provides the only mechanism of the decay of the hadrons with s-quark into the lighter hadrons without s-quark.
However because the weak interaction is, well, weak the hadrons with s-quark decay much slower than the hadrons of comparable mass but without s-quark. In the 50s of course no one knew about quarks and it was because of this surprising feauture that the hadrons with the s-quark were called "strange".