Accidental Hero: Believe it or not, there are actually a number of times when Gadget is legitimately helpful and actually contributes towards solving the case. It's just that he tends to do it purely by accident.
Repeated line whenever Gadget is congratulated for solving the case: I did?
Actor Allusion: "Would you believe I was practicing my swan dive?"
Flanderization: In the pilot, Gadget was bumbling, but not nearly as oblivious and managed to fend off most of MAD's schemes via competance. Early episodes occasionally flip flopped between this depiction and his iconic Accidental Hero persona before finally settling for the later. This was taken Up to Eleven where most spin offs such as Gadget And The Gadgetinis are concerned.
Flipping the Bird: Surprisingly in a children's series, he does this in the OPENING sequence! Granted, he wasn't trying to be deliberately offensive; his bird just happened to be a flashlight.
Papa Wolf: When he knows Penny is in danger, Gadget becomes virtually unrecognizable (and unstoppable) as he becomes the superhero he is supposed to be.
This also extends to other people Gadget cares about. He's gone to similar lengths to save Brain (when he actually knows that he's dealing with Brain, as opposed to thinking the disguised Brain is a MAD agent) and Professor Von Slickenstein, the scientist who gave him his gadgets.
Berserk Button: Just seeing Gadget, or even simply having to mention his name, can make Dr. Claw angry enough to smash holes in his desk or shatter diamonds with his bare hands.
Catch Phrase: "I'll get you next time, Gadget! Next time!"
Corrupt Corporate Executive: For the movie. Also any episode that revolved around MAD setting up a legitimate business and then wiping out their competition by some underhanded means, leaving Dr. Claw with a monopoly.
The Faceless: Not only faceless, but also bodiless as only his hands and arms are ever shown to the viewers. Luckily, in the movie, his whole appearance is RupertEverett.
Slouch of Villainy: Throughout the series, he only watches the events from his lair or car.
Surrounded by Idiots: Even with Penny always screwing up his plans, you sometimes think he might be ruling the world by now if his henchmen weren't all so stupid.
Villain Exit Stage Left: Whenever he's actually at the scene of a MAD activity, he'll typically escape in the MAD Mobile once the Gadget family ruins his scheme.
Non-Standard Character Design: In comparison to the other very cartoony designs in the show, Penny's is noticably more anime inspired and realistically proportioned. Gadget And The Gadgetinis gives her a somewhat wackier design, though still slightly more subtle than the rest of the cast.
Only One Name: Penny was never given a last name, though a lot of people just assume it's Gadget.
The Other Darrin: In the pilot she was voiced by Mona Marshall. In the second season, Holly Berger took over the role from Cree Summer.
Action Pet: Although more wacky than outright action, he still goes through things that one would have to be tough to get though.
Beleaguered Assistant: Occasionally averted when Gadget actually saves Brain. This usually happens when Gadget thinks he's a MAD agent and is trying to capture him, and ends up grabbing Brain right before he'd fall to his death.
My Instincts Are Showing: Though usually a neurotic Straight Man with more human intelligence than most of the police force combined, Brain sometimes has displays of puppy-like friskiness. For example, at one point when Gadget tries to throw away some explosives, he mistakes it for a stick and excitedly plays fetch with it.
Something Only They Would Say: Apparently, only the real Gadget would malfunction without warning and injure him with his gadgets. Quimby uses that fact to Spot The Impostor in one episode.