Hurley brought a guitar case aboard Flight 316, despite never being known to play guitar or own one prior to this. ("316") The case is similar in appearance to the one that Charlie brought aboard Flight 815.
He received the case from Jacob, who left it in the cab after his talk with Hurley the day before the flight. When Hurley alerted him that he was leaving his guitar upon exiting the cab, Jacob replied, "It's not my guitar." ("The Incident, Part 1")
Hurley attempts to stay afloat in the waterfall lagoon using the guitar case.
The guitar case survived the plane crash, as Hurley was using it to stay afloat in the waterfall lagoon. ("316") He later brought it along with him to the Barracks when he was inducted into the DHARMA Initiative as part of Sawyer's plan. ("Namaste")
The content of the guitar case.
The case also time-traveled with Hurley back to 2007. There, Jacob appeared to Hurley (following his death an hour before), and told him to bring the case with him to The Temple, where he needed to go to save Sayid. At The Temple, Hurley used the case to prove to the Others who had captured them that he had in fact spoken to Jacob. Dogen opened the case, revealing that it contained a large wooden ankh with a scroll hidden inside. According to Lennon, the message indicated that the Others needed to save Sayid. ("LA X, Part 1")