Continuity mistake: When Hannah makes the list with names involving her past events, they differ substantially between the close-up when they're written and the final wide angle once the paper is finished. Check Alexander: first an irregular oval with a straight side, then a perfect oval.
Other mistake: In the original version, a portrait of Bryce's family can be seen. Not only are the actors playing his parents completely different from the later episodes (which isn't necessarily an error, because actors for characters change in shows all the time), but Bryce is also shown to have a sister in the portrait. In later episodes, it is established that Bryce is an only child, so he doesn't have a sister. The portrait of his family is altered on the current streaming version on Netflix to show the new actors for Bryce's parents and remove the sister, but the DVD still has the original version.
Continuity mistake: During the first tape Hannah says that there are 13 tapes as there are 13 sides to every story, however we later find out that she only planned and recorded 12 tapes that night, and the 13th (Mr Porter's) was unplanned and recorded later on, so she would not have included it with the others when she mentioned how many tapes she was going to make in the beginning.