You can create a live star in the center sector of the dead star system. You need to build a ship with a star creator component. The odd thing is though that the dead star graphic will still be there, but with a live star at it's center.
To get rid of that graphic in game is a rather expensive process, but it can be done. You have to build a star there first. Then hit that new star with a stellar manipulation ship that will explode it into a nebulae or collapse it into a black hole. This will actually get rid of the dead star graphic. But then to make the system usable again you have to get rid of the nebulae or black hole you created, and make another star in the center.
Not really something someone would do unless they have wiped out all opposition and are just rebuilding the galaxy out of ego.
Geoschmo