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)
Revealing mistake: When Penny is launched out of the truck, she is supposed to be screaming, but it is apparent that she is smiling and laughing during the shot. (01:14:05)
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.
Other mistake: When Claw steals a whole bunch of gold, there seems to be a lot in there. First: How did he get so much gold in there with so little time? They seemed to have been throwing around the gold coins and stuff not to have enough time to do that. Second: Later in the movie you get an above look while they are trying to escape Inspector Gadget, there is less gold in the truck than there was earlier.
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.
Revealing mistake: When they freeze time towards the end, we see a boy jump into a pool and then the splash is in mid air. If you look closely at one point you can see him move a little underwater.
Plot hole: Why did Inspector Gadget put his glove on his right hand? We see a toilet brush in place of his hand with a glove on it, yet when he's flying through the air doused in toilet water, he has his glove back on.
Revealing mistake: Before the warehouse scene, Gadget and Gadgetmobile are talking and Gadget's hat illuminates with hearts. Look at the strap: after it illuminates it turns to a smaller cloth substance. It is let out that there's a computer in the strap used to make the hearts illuminate on the strap.
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.
Other mistake: When G2 applies lipstick after her torso rejoins with her legs, you can see that the lipstick never actually touches her lips.
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.