I bought one of the 360J's recently and prior to doing so I did a little research on what it was. While I am by no means an expert this what I found from past postings here on the forum and elsewhere on the internet.
The 360J is a .38 special version, sort of, of the 360PD. If you look at the posting of the picture where the lanyard is attached that looks to be a 360PD handgun due to the cylinder color being a lighter color. I believe the lanyard loop on those run along the back side of the grip, which is different than the 360J, where it is drilled right in the middle of the bottom of the grip.
The 360J, from what I read, is produced by Smith and Wesson for the Japanese Police. They get released into the US market when there is an overrun or for some reason the contract is canceled. When I purchased mine recently, and the price was to good to say no, the vendor said they were from an order that was canceled. And there did/does seem to be a bunch of them available right now.
The lanyard hole in the bottom is used with some special grips, that I have no idea where they come from, and a lanyard knob attached at the bottom. I assume this is for the officer to retain their weapon while on patrol.
If you Google Smith and Wesson 360J Japanese Police there is a picture from an AR15 Forum site that shows these grips and the lanyard that goes with them. The poster there is asking about where to obtain the grips, and I must say if I could find them and that set up I would probably pick up some as well, and posted that S&W wouldn't tell him where the grips come from.
At one time when I Googled this gun I found several sites from Japan that appeared to sell this gun with the grips and lanyard. I found that weird because I believed Japan had pretty strict gun laws. Then when I looked deeper into the sites they were replica guns of the ones carried by the Police.
Smith and Wesson probably puts the longer grips on them here to hide the lanyard hole for the US market. I picked up some Bianchi Lightning grips for a J Frame and after a little bit of custom cutting put those on it.
Not sure how to fill the hole or if you could. There is no pin with these guns.
Sorry if a bit long winded.
Good luck and hope this helps.