coaster brake lube?
 

I'm guessing this coaster hub will work a bit better without the string.
What kind of lube should I use, and how much of it?

http://i.imgur.com/3KzIMeL.jpg

http://i.imgur.com/dsSCaJv.jpg

Just about any type of medium body grease is fine - those hubs are reasonably protected from the elements. Be generous but don't pack the interior with grease. the bearing cages and cups should have enough to resist intrusion of water. The brake shoes get grease, too - don't make the mistake of thinking you should keep them dry.

If you live in a hilly area coaster brakes can get pretty hot. For that reason, I use an automotive grease rated for use on the disc brake front hubs (auto). This id formulated to handle the heat without melting or burning.

Nice job cleaning it up. I did a couple coaster brakes last fall, pretty interesting.