Trivia events

Trivia events can be run alongside lesser attention-demanding events like mining shared cans or as standalone events.

Collect a list of EVE-related trivia. You can follow a theme if you want: lore-related, faction-related, ships-related, etc.
Decide on a point system if you're planning on awarding prizes.
If there are enough participants, you can split them into teams. Set things up for a team event beforehand (Mumble comms, a different point system, prizes).

For Mumble: if there are teams, it's a good idea to use one of the fleets channels so that each team can use the subchannels for private comms. If not, make an on-demand Mumble channel that you can customise.
For chat: the Events (EVE Uni) is open to use for events like this. Teams can also use group chats to talk among team members.

Set up teams if you're planning on organising the event that way. Ensure that each team has its communications set up.
Teams should also decide on an order in which to take questions: Pilot A goes first, Pilot B goes second, and so on.
Before you ask each question, ensure that the player is not AFK and can respond. You can do this by asking them to verbally confirm their presence on Mumble, or x-up in chat.
You can use a buzzer system if there are teams: the first player to x-up in fleet chat after a question has been asked gets to answer it.