DIAGNOSIS That's baked-on grease and carbon, a normal byproduct of grilling. The porcelain-enameled lid is durable, but needs the right technique to clean without scratching it.
FIX You want to soften the deposits first. While the grill is cold, spray the inside of the lid generously with a grill cleaner or a paste of baking soda and water. Let it sit for 20-30 minutes. For tough spots, ball up some aluminum foil—the scrunched foil is abrasive enough to break up carbon but gentle enough not to damage the porcelain. Scrub in a circular motion. Wipe clean with a damp cloth. For maintenance, after each cook while the grill is still warm (not scorching hot), wipe the lid with a damp, folded paper towel held with tongs.
IF THAT DIDN'T WORK For extreme, neglected buildup, you can use a plastic putty knife or a nylon scraper (like for cooktops) to gently lift the thickest layers off first, then follow with the foil scrub method.
HEADS UP Avoid using metal brushes or steel wool on the porcelain lid—they can leave tiny scratches that will actually make future cleaning harder and can compromise the finish. Also, never use oven cleaner spray, as it can be too harsh for the materials.