The Eon Ticket is an item that has been initially distributed to small groups of people - relatively speaking - through a number of means; for the most part, the distributions have already stopped, so don’t bother too much about looking up how to get it.
Back in the days of the original Ruby/Sapphire, the Ticket could be transferred between the two via record mixing. There is another method of transference for Omega Ruby and Alpha Sapphire: StreetPass will transfer the ticket. Keep in mind that you will not lose the ticket when transferred!
Therefore, you only help people by StreetPassing it, or HomePassing it, even. We won’t go into details about what HomePass is, but it basically involves creating a fake StreetPass Relay in the comfort of your own home and then cycling the MAC address to connect to other possible relays.
Purpose of the Eon Ticket
But what’s the point of the ticket? The Eon Ticket, much as in Ruby/Sapphire, allows you to capture the Lati Pokémon that is opposite your version. Already you can get its Mega Stone from your mother in Littleroot, but without the Pokémon itself, there’s not much point.
Granted, trades are always possible - and, in fact, far more possible than in Ruby/Sapphire - to get the other Lati, but we would guess Game Freak included it for the nostalgia.
Another benefit is the one you wouldn’t expect; the Soul Dew can be earned from this. It is a hold item for both Latios and Latias that boosts their Sp. Atk. and Sp. Def. by 50% each; so much so that it had to be banned from the Battle Maison and the like!
Using the Ticket
Note : To obtain the Eon Ticket, you must have acquired the BuzzNav app and activated StreetPass from the app. Then whenever you get a new notification in your BuzzNav, keep checking it until you eventually get the ticket from somebody (could be never…).
In any case, once you obtain the ticket, go speak with Norman in the Petalburg Gym. After doing so and listening to his spiel, you’ll be able to take the ferry to Southern Island.
There, you will encounter Latias (in Omega Ruby) or Latios (in Alpha Sapphire) at Level 30, much as with the one you already got. After the battle, be sure to pick up the Soul Dew from the ground! It is a very valuable item, so don’t miss out on it!