After you beat the DLC and go into Alija's study area, there's a holotape on his desk titled for Veronica. How do I give this to her?

Ask her about her relationship with Elijah.
Ask her if she ever found him, where he went.

Exit conversation.
Talk to her again.

There will be a new dialogue option. "I Found Elijah".

You can take it from there.

I tried just about everything to get the dialog option to appear, both inside and outside the abandoned BoS bunker.

Then I miraculously picked up father E's robe out of the BoS chest at the foot of the bed in the abandoned bunker and the option appeared when talking to veronica.

Choose the "Tell me about yourself" option and talk to her about Father Elijah. It will then appear on the "main menu" of her dialogue.