Dmitri Mendeleev’s periodical table brought order to the chemical elements. It did not explain this order, but it was an enormous step forward for chemistry.
Likewise, Murray Gell-Mann and Yuval Ne’eman (independently) brought order to baryons and mesons (all the particles that “feel” the strong nuclear force) in 1962 by proposing that they fit into a mathematical structure known as SU(3). Gell-Mann called his proposal the Eightfold Way (which he came to regret later on when New Agers seized on imagined ties between eastern religion and philosophy and particle physics). In 1964 a particle with properties predicted by the model was discovered, which lent it much credibility.
So there was a pattern, but what the reason behind it? There was a fundamental representation of SU(3) that did not have particles assigned to it in the Eightfold Way. If particles were assigned to this representation and they existed, one could hypothesize that all the other baryons and mesons were simply built up from combinations of these, like atomic nuclei are built up from protons and neutrons. But there were three major problems that prevented Gell-Mann and others from jumping immediately on this attractive proposition.
(1) Fractional electric charge. The math dictated these hypothetical particles have electrical charges of +⅔ and -⅓, which had never been observed, and seemed outright crazy to most physicists at the time. Much time and effort went into complicated schemes trying to circumvent this.
(2) Statistics. Bringing these particles very closely together would violate the Pauli exclusion principle.
(3) Freedom vs. confinement. If these particles could roam free, then why had fractional charges never been observed? If they were always confined to composite structures with integral charge, then that would imply strange properties of the strong nuclear force, and it seemed doubtful that a local gauge field theory with massive bosons (force carrying particles) could even have such properties.
By 1964, Gell-Mann felt confident enough to predict that these particles were indeed “real” despite these issues. He called them quarks. (Independently and at the same time, George Zwieg was led along a different track of reasoning to propose the same thing, he called them “aces”.)
The statistics problem was solved by introducing “color” (see the link provided by @ETpro) as a new quantum number. And asymmotic freedom was discovered in massive local gauge field theories, allowing for confinement (and explaining why fractional charges had not been observed). This led to a complete theory of the strong force: quantum chromodynamics (QCD). QCD has been on the experimental rack since the 1970s. It is very hard to precisely calculate things with this theory, but it has survived all the experimental tests thrown at it so far.