One thing that struck me about the Lincoln Museum is the library attached to it. One section was a small private collection of books about all the presidents (“Someone wrote a book about Millard Fillmore?” Yep.) It has convinced me that any real White House Museum also needs a research library—broader, but not as deep as any of the existing presidential libraries.
Ooh, I just thought of something. The library could be built as a replica of the Taft-Hoover West Wing, with a replica of the original Oval Office as the centerpiece.