What does best among the worst mean? For example, if a person is best among the worst does it mean that he is the worst of all or that he is a good person in a group of bad persons?
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The best among the worst literally means what it says. If a person is the best among the worst it means he has the most good qualities among the group of people with the least good qualities.
Your doubt is very understandable. It is easy to interpret the phrase as 'superlatively worst' as you have, though incorrectly, suspected.
Here, worst is not an adjective but a noun and therefore could take best as an enhancer. However, it does not. So phrase does not mean 'superlatively worst'.
It simply means that he is the best among the group of people, the group being the worst of all.
I would say that best among the worst probably means least worst in most cases. However, it could be taken to mean most worst (i.e. best at being bad). As such, I would recommend against the use of this phrase in any formal or business writing, to avoid any potentially upsetting misinterpretations.
If someone else uses this phrase, you should work out the meaning from context, because there is some scope for both meanings.
Nonetheless, without any context to go on, I would say it probably means "Of a group of bad people, he is the least bad".
A more common phrase is "least worst". For example:
That is, all ways to stay fit are unappealing, but since running is the least unappealing, that's the one I do.
"Best of the worst" is (probably deliberately) vague, because it says "there is some set of bad people that includes this person, and the worst person in the world, with this person being the best among them." -- and the size of that group is unstated.
Perhaps it is clearer if I say "Nine is the largest of the smallest integer numbers". Well, it is if that group of "smallest integer numbers" is 0-9.