DIAGNOSIS Grease and food spatter get forced into the narrow gap around the hinge when you close the lid, and then it gets cooked on. The design makes it nearly impossible to prevent and very difficult to clean.
FIX The best method is preventative: after cooking and before the grease hardens, wipe the hinge area with a damp paper towel while the grill is still warm (but not scorching hot). For baked-on gunk, unplug the grill, let it cool completely, and use a soft-bristled brush (like a clean toothbrush) dipped in warm, soapy water to scrub the crevices. You can try a plastic pick to gently dislodge chunks.
IF THAT DIDN'T WORK For stubborn residue, make a paste of baking soda and water, apply it to the hinge area, let it sit for 15-20 minutes, then scrub with the brush. Avoid spraying liquids or submerging the hinge, as water can get inside the housing and cause electrical damage.
HEADS UP Never use metal scrapers or steel wool around the hinge, as you can scratch the housing and make future cleaning even harder. Also, never run water directly over the top hinge assembly; you'll short out the electronics in the lid.