Most rivets can be drilled out. I have not done this on a door handle on a Mustang but have done at work lots of times (heads up! I don't work with cars in my job). If you have the replacement rivets, look at the small diameter of the hollow piece. Either use a caliper and find the right size bit or hold drill bits behind the replacement rivet up to light to see if you have close to same size. Larger will be better. Another way that works smoother but you must be carful not to go through, is to use a bit where about half the bit will fit the hole on the hollow end and drill until you have removed enough material to cut the hollow part so you end up with a washer looking thing on you bit and the rivet will push through. Don't keep drilling once you think you have that washer-like thing. Being that the bit is pointed you will gradually remove material until your drill point gets deep enough to match the size of the rivet body and the hole drilled in the door's sheet metal. If the rivet spins with the bit, go at a harsh angle with a lot of pressure (not straight but at about 45 degrees, hold that angle and rotate counter-clockwise) but beware that once you break through, the sheet metal may draw the bit right through. Also, if it spins, use a vise grip or plier (needle nose vise grips are my favorite) to hold the big end. This probably has you all confused but e-mail me and brandkoe@yahoo.ca if you want more clarification. Good luck.
Done it. What the other poster said. Will be drilling at angle. Get a sharp new bit. Remember you are not in a hurry. Hardest part is popping it back. Maybe a good idea to tape a cloth around dood handle to protect the paint of the door. When riveter pops you got pressure on it. Hand and riveter wants to jerk. May hit the painted part of door. Hope that made sense.
BrandKoe I notice someone else like Ragtops. I have a MG Midget, building a MGB with 2.3 turbo. I tried once to buy a Miata but I tried when they were first introduced. They wanted way over sticker price.
you can drill them from outside the car just be carfull not to scratch your paint if you have the plastic handles just pull up on them until they break off your going to be replacing them anyways then you should be able to get at the rivits better drill through both of them and the rest of the handle will come out after you disconnect it from the pipe running to it.
Door Handle replacement.. Rivet Removal, 1989 Mustang 
