Creaking rocking chair

Depends where the noise is coming from. If the spring doesn't have parts that rub, chances are that isn't the noise source. Looking at the photo, the wooden parts seem to 'roll' over each other. Grease probably isn't good on wood. Talc powder is an old fashioned cure under creaking floor boards, you could try that or French Chalk.

It doesn't look like it's the original bolt - if it's under-size it might be moving in a hole where it should be snug. There's probably a metal insert between the spring and the wood, greasing that might help.

You use the word "creak", metal on metal tends to "squeak". You really need to track down where the vibration/stiction causing the audible creak is orginating. This may not be the same, or near, the place the sound appears to coem from. If you have a nice long screwdriver use it as a probe to find the real source. Place the end of the handle against an ear and keep it there whilst you touch various parts of the chair and listen as some one reproduces the creak. When you find it it wil be loud and clear, further away quieter, muffled or inaudible via the probe.

Or silicone furniture polish on the rocking surfaces.

Beeswax is the traditional lubricant for wooden drawer runners. It is probably more persistent than a powder.

Are you sure its the mechanism itself and not part of the support structure itself? Brian