Take a look at the connector that is on VHF radio side. It should be the same connector. From the photo you have provided it appears to be an unusual setup.
You would have to pull the coax up about 4" to install the new connector, you also need soldering iron.
Haro

This is a standard connector used in all VHF installations. It is called a PL259.

If you are going to take the effort to splice on a new PL259 connector in a bosun's chair, you mine as well run a new line. My radio reception improved 5x after running all new line and antenna. I used a 100' fish tape from harbor freight and inserted it into bottom of conduit inside mast. Used a cheap $40 endoscope camera to see what I was doing. Pushed it all the way up until it hit the top mast plate and reached the 55' mark I had pre made on the fish tape. I drilled a hole on top of mast and used endo cam with attached clip to grap end of fish tape and the pink string loop I added. Pulled it out of hole. Taped new cable and then had to have a helper pull tape and cable down mast- through the conduit and out hole in bottom of mast. You need a 1" hole at the bottom of the mast that faces into the cabin. Some boats have it on the ends or facing the wall. Photo on left is the top of the mast, looking into the hole I drilled. You can see the pink string loop that I needed to grab with the endocam clip and half of the top of the conduit faintly in the distance. Photo on right is the base of the mast where conduit ends.