It is presumably a play on the English word 'grotty meaning 'seedy'. Grotti's logo caricatures Ferrari's; in GTA San Andreas, the galloping horse is replaced with a rabbit/hare and, in GTA IV, instead of a galloping horse, the logo depicts a horse sitting down.
The Grotti Showroom in Liberty City is one of the many buildings that allows you to get to six stars wanted level quickly. Just get to the first floor and start shooting. You can even hold the people inside the building hostage by blocking the front door with a car.
If you enter the Grotti Dealership in Middle Park East and use a Baseball Bat to knock out all the windows on the main floor, the windows on the second floor will begin to fall either on their own or if the player walks into them.
The doors in the Algonquin showroom will occasionally lock but the windows still manage to break.
It is shown to be a very prominent car manufactuer, as shown when companies like ElitasTravel.com call themselves the "Grotti of air travel", similar to how some companies call themselves the "Ferrari of" their work.
The Grotti emblem in GTA San Andreas has a rabbit in a position resembling the horse's position in Ferrari's emblem, while in GTA IV it features a horse that's sitting down, again parodying Ferrari. GTA V introduces yet another logo, depicting four arrows arranged into a four-leaf clover on a blue shield.
The rabbit logo of the 3D Universe rendition of Grotti is reused in the HD Universe as a logo for the Ushero brand of clothing.