Hello, Nazryt,
The part you reference is called a 'pawl', and its function is to hold the ratchet disc, which, with the chainwheel and the brake disc, forms the brake assembly which holds the load. The hole in the pawl is an oil hole for lubricating the pawl, so that it will not 'stick' on the pin which holds it.
Since there are really obvious safety considerations with any sort of overhead lifting equipment, I would strongly suggest that you have the chain fall completely apart, carefully clean (solvent and fine-bristle wire-wheel, not electrolysis) all its parts, then very carefully inspect for any cracks or other signs of damage or abuse.
A chain fall is a simple device, so disassembly, cleaning, and re-assembly will be obvious, but it can't hurt to take photos or make sketches as it comes apart. Clean and re-grease the load shaft bearings, and use some 'sticky gear lube' on the gears.
Clean the leather brake disc carefully. I would just scrape the build-up of grime from the surface of the leather, instead of risking the use of any solvent which could attack the leather.
If you know anyone who has a deburring or finishing tumbler, getting the hand and load chains tumble-cleaned will make them feel 'as new'.
added on edit... I've 'serviced' or 're-conditioned', i.e. just clean, inspect, and re-lubricate, a few of this older type of chain fall, one time and another. It really is a quick, simple, and easy project.
cheers
Carla