Would it likely be just an accident from getting dust on some wood and it starting faster, or something like a witch-doctor member of the tribe who experimented all the time? I know I'm asking for a good deal of speculation, but hopefully this still fits within the rules of the sub.
This would be a good question for /r/askanthropology. That said, there is a false assumption that one often encounters in the questions in this subreddit - that is, that people in the past were "primitive," meaning ignorant and not sophisticated. This is a false perception that leads some of us to wonder how people in the past could accomplish remarkable things.
The fact is, hunter gatherers were keenly aware of their environment and they understood what various resources could yield. It is not difficult to imagine that someone could have discovered the properties of a manganese deposit, and knowledge is easily passed on. We cannot know (and no one at /r/askanthropology can really know either) specifically how this discovery was made. But people are inquisitive and constantly testing the bounds of their environment. The recent discovery of the use of manganese by Neanderthals simply gives expression to this point.