Hi Kate & Roh - thanks for the comments!
Thorongil probably wouldn't give her a book as he doesn't know that she knows about Elrond, he doesn't know about the book she and Denethor looked at, and he's not coming back to Gondor while she's alive once he leaves after Umbar. However, I can assure you that he does give her a few things, most importantly his trust. Though you've given me an idea for a future plot point...
There's a bit more to know about Thorongil's true identity, though it is mostly filling in the edges. He is the heir of Isildur, as shown by the Ring of Barahir, and that's all they really need. It does matter how he is the heir, whether by male or female line descent, as the Steward's line (in this story) does have a distaff relationship to the line of Anarion. This raises questions about who has the greater claim. My guess is that someone with direct descent from both Isildur and Anarion would out-rank someone with descent only from Anarion, even if they are distaff relationships.
Denethor would be more respectful of Elrond than of Gandalf, if only because he is a blood relative, but I don't think he'd care much for any outsider poking a nose into the affairs of Gondor.
If writing goes well, I should have a new chapter out this weekend that presents more information about Thorongil, and what Denethor can puzzle out from it.
Toodles - Ang