That's a tough question. Not because the book is boring, but because it is so full of exciting moments.
Billy dies, and young Jim searches through his stuff and finds some documents. That's pretty exciting because you, the reader, get a feeling that those documents are important. That's confirmed when Jim asks some men around the town about the documents and the map. Jim is told that the map leads to a huge pirate treasure. Some books can be sort of boring in the beginning until a "call to action" arises for the hero. From that moment forward, the story is propelled along at an accelerated rate. For me, that call to action is always an exciting moment of the story. Like when Obi-Wan asks Luke Skywalker to come with him. It's not action packed, but every time I watch it, it's exciting because I know that it is a critical moment. It's the same in "Treasure Island" with finding that map and learning its significance.
Battles and fights are always great to read about too. The first battle between Silver, his mutineers and Captain Smollett is definitely exciting. The tension is heightened too with Smollett being wounded.
Probably the most exciting though is not a single event but a series of interconnected events that propel the reader on quite quickly. It's the climax of the story, which is what Stephenson intended. Silver is struggling to handle his own mutiny, Jim is begrudgingly helping Silver, the treasure location is found to be empty, and a final battle between Smollett and the pirates. So great! Then when the reader is thinking that the treasure is gone forever, Ben reveals that he had found it months earlier and moved it to a hidden cave. Victory for the good guys.