Prior to the last Olympics, Nate Silver, then of the New York Times, did a study on which Olympic sport is the easiest to medal in. This is not to say that winning any Olympic medal is easy, but some sports have a high barrier to entry (equestrian), a very competitive top (beach volleyball), or award so few medals that it makes little sense to try (basketball).
That being said, there are some sports that spread their medals around. When you factor in the number awarded and the national distribution of medalists, it's the individual combat sports that rise to the top: wrestling, taekwondo, boxing, and judo. (The fact that those four have numerous weight classes and award two bronze medals in each does not hurt.)
Take Israel, for instance. It's not exactly a world athletic power by any stretch of imagination, but in Rio, it added two bronzes (both in judo, Yarden Gerbi and Or Sasson) to bring its all-time total to nine. Five of those have been in judo. Clearly, the Israelis have found a niche.
So, poor medalless countries (Albania, Belize, Chad, etc, etc, etc): stick to the right sport, and you have a chance! Just keep away from judo... that's Israel's thing.