I don't often fit them.

While there is an increased risk of grit getting into the seal and opening it up, the rubber bellows are often not the OE spec and can cause a car to jump out of second, fourth, or reverse, if it's too thick or long - the compressed rubber pushes back on the linkage and off-throttle can cause it to de-select the gear.

Edit: the bellows perform no oil sealing function, they are merely a dust cover.

Is your mini leaking oil at this spot? If it is not, then leave it alone. This shift gaiter boot is rubber crap and will deteriorate again. Wait until it starts to leak and  then replace the seal as willie b. says.

This is the kit that has the parts you need.

https://www.minimania.com/part/MSSK050/Classic-Austin-Mini-Rod-Change-Anti-Oil-Leak-And-Seal-Kit

Put the car in reverse and drive the roll pin out of the coupler, the one closer to the trans. You will need this tool to drive the roll pin out. Do not use use a regular center punch as it will flare the end of the roll pin and jam it in the hole. You do not (and don't want to) drive it all the way out of the outer part of the coupler. Should be able to use a large pair of channellocks to squeeze it back in to place.

https://www.minimania.com/part/TOOL17/Gearbox-Selector-Roll-Pin-Punch-Tool

If you want to replace the aluminum part inside that has an "O ring" on it you might be able to pry it out or instead drill two holes in and put in small screws that you use to pull it out. I have read where some folks just knock it in further and then put the new one in behind it. Be aware you will to drain the engine oil before you do this.

Does any one know what this piece of rubber is called and how I can replace it? I went through my entire car the other day to check every single piece of rubber that I have to replace (it's ridiculous) and I found this between the transmission and the rod that goes to the stick shift it's self.

Thank You!