Continuity mistake: In some printings of the book, in Chapter 2, when the Dwarves have to camp during the rain, it is stated, "So far they had not camped so far on this journey." This is a contradiction to an earlier paragraph which says that they had camped earlier. This is due to the earlier paragraph mentioning them staying at inns in the first edition, which was changed in the third edition to them camping, but the latter paragraph was not changed. Some printings do correct this however, but not all.
Factual error: The book mentions Bilbo's sword as being originally made as a dagger for an elf. The problem is, proportional to the wearer, that doesn't work, weapons and tools have to be made with the user's exact size always in mind. The blade or handle would be too thick, or something else would be off. I know this from trying to use a figurine sword to open envelopes.
Suggested correction: The origins of Sting are largely unknown. Even though it was made by elves doesn't mean that it was meant for elves. The book doesn't say it was a dagger for elves, just the size was comparable to one. It's only said that it was very small for Elf standards and that means it has a small handle perfect for a Hobbit hand.