Being a Systema it is probably pressed in but who knows for sure?
I have a Systema pistol and several mainspring housings, the pin is not cup pointed like most Colt and I will bet money the steel isn't quite up to the Colt standards either.
I have removed quite a few lanyard rings for just the reason you wish to and have not yet had to resort to an arbor press to get a pin out.
Jim Keenans starter punch suggestion is a good one.
You can secure the heated housing in a good quality vise between two vise jaw pads made of 1/16" thick brass sheet to prevent the jaws from marring the housing.
Use a welders glove or good pliers to hold the heated part while securing it in the vise
As for drilling, I use half length jobber drills for this kind of work and trying to center punch a 1/16" or 3/32" pin is well,,,trying.
Use a drill press and high speed steel bits. You will need at least 600 RPM on the spindle. 800 rpm would be better if you can get it.
Secure the part well in the drill press vise and make darn sure the pin is 90 degrees to the drill bit, use a bench level.
Peck feed the drill bit into the pin/hole, drill a little bit at a time and withdraw the bit and clean by hitting the rotating bit with an acid brush dipped in drill oil, this will keep pin chips from building up in the new hole and then cutting the hole out of round or oversize.
More advise for the vise and punch method.
You support the area to be punched above and resting on a pin hole of a bench block that has a wood block underneath to bring it to the height plane of the part.
The securing pins vary in diameter from 1/16" to 3/32" in diameter.
If you redrill the hole you can go up to 5/32" in pin and hole diameter but it will really hog out the lanyard pin slots.
Green #609 Loctite retaining compound placed on the lanyard ring posts and on the new pin will ensure the lanyard will not jiggle and the new pin will not drift out if you get sloppy. HTH