Continuity mistake: After Gadget and Penny read Claw's letter, look at Gadget's hat, the folds change from crumpled to straight. (00:28:50)
Continuity mistake: When you see the truck at the Federal Reserve, the truck has no wall between the front and back, there are several stacks of gold bars and coin sacks. But when it shows the overhead view as Claw's rocket comes out, there is a wall between the front and back, there are no sacks of coins, the gold bars are much smaller and the stacks of gold bars take up less space than before. (01:15:55 - 01:24:55)
Continuity mistake: At the end of the movie, when Inspector Gadget gives an award to Penny he puts it around her neck and when the camera faces her, her hair is in the ribbon, but when they show her back it is out of the ribbon and then repeats itself about 3 times.
Continuity mistake: When Penny is strapped to the cart, Inspector Gadget frees her and they both run away from the cart. The cart then explodes and Inspector Gadget is propelled forward. The Gadgetmobile pulls up and Inspector Gadget lands on the bonnet from a position directly above, which would be impossible if he was going forward.
Continuity mistake: In the scene on the bridge where Claw's Ice Cream Truck is coming toward Inspector Gadget, G2, Penny, and the Gadget mobile, the first shot shows the truck not far away. It is driving toward them, but takes a long time to get there. The truck takes about 15 or 20 seconds to reach them, although it should have taken less than five, according to the first shot.
Continuity mistake: When Gadget gets thrown through the Men's Room door, it smashes to pieces and his toilet brush attachment is replaced by a rubber glove. On the newspaper front page, it now shows the Men's Room door is no longer destroyed, and the toilet brush attachment is now on his hand instead of the rubber glove.
Answer: Probably just a common trope seen in movies to indicate something is on or working. It adds a hint of in-movie realism. While it's shown that the claw can be operated manually or through will, it isn't implied the light does anything or tells us anything.