Elijah obviously intended this to go to Veronica, but you earned this, right?”
To obtain this perk, after completing the quest line in Dead Money, leave the Sierra Madre area. When you reach the Abandoned Brotherhood of Steel bunker, turn right into the side room. Use the terminal here and choose the Download Holomessage option. Head back to the Mojave Wasteland and find Veronica. Initiate dialogue, choose "Tell me about yourself.", then exhaust all the options relating to Father Elijah.
Now back out to her main dialogue menu and choose "I found Elijah". To get this perk, you must ask her to unlock the message for you, then refuse to give it to her. Choosing to let her have it will instead give Veronica the Elijah's Last Words perk.