Okay, I found out how to finish Dead Money successfully, and I got Elijahs note... But... I can't give it to Veronica. What do I do?

Go through her dialogue tree's again. (either in the tell me about yourself or ask questions one, cant remember.)

First, exhaust all dialogue options regarding Father Elijah. Then, find his note in your inventory and select/click on it. This doesn't actually do anything, but the "I found Elijah" dialogue option should now be available when you talk to her.