I do agree with you that we don't really need anyone to recommend the Iliad or To Kill a Mockingbird for the next generation. Among books handed to me, well, if we're defining that loosely Larrys choice is good for all the reasons he stated, but I'm not sure a book like that can really count as being "handed to" anyone unless they remember who handed it to them; it's part of the Wests cultural heritage. I'll go with The Adentures of Tom Sawyer as a snapshot of the antebellum South; Huck Finn is undoubtedly more popular and probably deeper, but a lot less fun to read, plus there are plenty of other exposes of slavery (such as the one referenced below). Unfortunately, most books from my lifetime that I've read I wouldn't recommend.

Among books published in my lifetime I'd likely say All the Presidents Men or Roots--but that's on relevance and reputation; I've not read them.

I'd recommend All the Kings Men (the similar subject of which probably contributed to the title of Woodward and Bernsteins non-fiction work). It WAS handed to me (though not published in my lifetime) and remains as relevant an illustration of the dangers of populism turning to demagogeury as it was when published--ten years after my father was born.

The Lord of the Rings was handed to me and would probably be my choice if I could only make one--but it also wasn't published within my lifetime, and isn't a single book.

The Sprawl Trilogy was published in my lifetime, handed to me and would be an excellent choice, since it's well written, many of the issues that were almost entirely fictional when it was published have since become facts of life and more of them almost inevitably will in the near future--but it's a three volume series, not a single book.

Among individual books I've read and that were published in my lifetime, perhaps The Mists of Avalon; it offers more of the "women's experience" so well represented at your link, as well as the fantasy genre revitalized in the Twentieth Century.