Welcome to our little corner of the web. Honestly, I'm not sure about the little button. But, if you remove the two screws that hold the oscillator box on (one of the screws is visible in your photo), you should be able to remove the old semi-solid grease and replace with new.
You may know this already, but those Eskimo oscillating mechanisms are not very high quality and are usually the first component of those fans to fail. Otherwise, they are cute fans and of decent quality for what are known as "dime store fans." (Less expensive fans sold at discount stores) However, these all-metal fans are much better than any new plastic fan imported from China.