Wil Wheaton and Patrick Stewart came up with the gag where Wesley Crusher encounters a non-automatic door for the first time. They approach the door, which Wesley stares at. Picard looks at Wesley, as if thinking "Aw kids!" and opens it for him.
The original plan was for the Captain's yacht to be used to transport Capt. Picard to Starbase 515 for immediate heart surgery, but the producers decided to nix the idea due to budget constraints, and to go with a traditional shuttlecraft instead. Ironically, the Captain's yacht for the Enterprise-D is never used nor shown throughout the entire Star Trek: The Next Generation (1987) series run, however the one for the Enterprise-E would be used in Star Trek: Insurrection (1998).
The trivia item below may give away important plot points.
In the final scene when Captain Picard states "any rumors of my brush with death are greatly exaggerated ", he is paraphrasing a quote attributed to Mark Twain that goes "The rumors of my death are greatly exaggerated".