Just try a "cleaner, paste" type wax and lightly clean it up. It's old and will never be like new, so appreciate the patina. Use a "Q" tip around the edges. The first thing though, is to learn the correct way to remove a badge without marring up its face . It's covered here in this thread. My personal favorite way is to remove the bike's fork and take a 3/8ths ratchet extension bar and use the female end to push the underside of the rivet ends up. You can pry them up a tiny bit. It makes them easy to grip. Use quality electrician's blunt nosed pliers to grip them and twist the rivet counter clockwise and pull it out. The face of the badge should always, and I mean always be covered in painter's blue, easy-off tape prior to removing or installing, to prevent nicking the face of your badge. Below ,I illustrate the proper way to remove a head badge.
Tools you will need to safely remove your badge.
First, carefully cut the head tube off your bike.
Mount head tube in vice. [You don't want to damage it.]
Select a fine bladed screwdriver. [ Very important to use your best screwdriver]
Carefully remove the badge. Do not "rush" the job. This point is critical!
Pry up. [carefully!]
Last, but not least, lightly clean with your favorite paste wax. And your'e done!
This is a fun hobby. I hope you enjoy it as much as the rest of us do!
