Here is some better details about the eon ticket.
The Eon Ticket is two things. First it is an item that can be download into either Pokemon Ruby or Sapphire Version it is also a Pokemon Battle-e Card. This item allows the player to travel on the ferry to a special island in the Hoenn region simply called "Southern Island." There you have a one time chance to catch the legendary Pokemon Latias (Ruby Version) or Latios (Sapphire Version).
The Eon Ticket can be downloaded in three ways. First a player may obtain an Eon Ticket from Nintendo through a link to a special game pak that Nintendo uses. Second you can use the Eon Ticket Pokemon Battle-e Card for the E-Reader and transfer the eon ticket via Game Link Advance to your Version. The third way to obtain the item is to receive it by mixing records with a person that has a eon ticket in his or her Ruby or Sapphire Game Pak.
Remember to receive the Eon Ticket you must have MYSTERY EVENTS activated in your Pokemon Game Pak. To do this first you must obtain at least five badges (however you must beat the elite four and the champion to use the Eon Ticket). Then go to Petalburg City and speak to the person standing next to the computer and change your trainer profile message to MYSTERY EVENT IS EXCITING. With that Mystery Events will be unlocked and you can start using special link ups such as Pokemon Battle-e Cards.
Using the Eon Ticket
Unless you obtain the Eon Ticket by Mixing Records, you must go see your "dad" (the Petalburg City Gym Leader, Norman) and he will give you the Eon Ticket.Then you can take it to Lilycove City and talk to the Hostess at the boat entrance. BEFORE TALKING TO THE HOSTESS, SAVE YOUR GAME! After the Hostess and the Sailor talks to you, you'll find yourself at Southern Island. DO NOT SAVE AT ANY TIME DURING THIS POINT. Although the island is filled with trees, you will only find a cryptic saying plus a monument in the forest center of the island. By walking up to the monument and pushing the A button. A Latias or Latios will appear and attack. Like many other legendary Pokemon, this is a one time shot to catch the pokemon. Once you have caught the pokemon, another message will be display (which I haven't the foggiest what it means) and there will be nothing left to see on the island. If you fail to capture the Pokemon you can restart without saving and try again. When you leave the island there will be no possible way to return to the island so don't save until you have successfully caught the Pokemon.
Tips for Catching the Pokemon
Latias is easier to catch than Latios for a couple of reasons. First I thought using Ice Beam to potientally freeze the Pokemon would work, well with Latios (not Latias) it doesn't. Latios at LV.50 has the attack Dragon Dance which automatically defrosts when used. So only sleep techniques will work there. Also Latios has more powerful Psychic type moves than Latias. Try to keep the pokemon asleep as much as possible because both have the move recover (which restores half of its maximum HP). Other than that, just do what you do best. Catch em all.
Notes about Mixing Records: If you have a Eon Ticket, you can give it to other players by Mixing Records. This can be done 100 times. The player receiving must have the grey link of the GBA link cable inserted into his/her game boy. A player with a Mixed Record Eon Ticket cannot pass the Eon Ticket him/herself the same way.
About Enchancers (such as Game Shark): It is dangerous to mess with these things anyways but there is a likely chance that Eon Ticket receiving won't work with the Eon Ticket. Overall, BE A TRUE TRAINER AND THROW THAT GAME SHARK IN THE TRASH!
The only way to obtain a new, usable Eon Ticket is by restarting the game.
Well that is it. Hope that helps.