Big event mystery boxes are rewards from the Birth by Fire and Build A Beach events. It can be opened for a random selection of rewards, and it is destroyed upon opening.
Birth by Fire
During the Birth by Fire event, two of these boxes are awarded for turning-in 8,500 burning embers to the Fallen Nihil. Alternatively, 500 burning embers can be directly converted into a big mystery box (at the expense of not progressing further in the event, however).
It can be opened for a random selection of five rewards, and it is destroyed upon opening. Five empty inventory spaces are required to open it.
Three of these thematic rewards may be obtained by opening a big event mystery box throughout the duration of the event.