Lincoln's diary is found in the Museum of History Lower Halls. It is in the large display room with the door that leads to the Museum of History Offices. The diary is lying on the first display on the right side of the room.
Related quest
Lincoln's Profit Margins - can be sold to Leroy Walker, Hannibal Hamlin, or Abraham Washington.
Notes
Though it does not appear in the game, this item is found in the game files for Fallout: New Vegas.