When I first saw it in Great Mambo Chicken and the Transhuman Condition, it followed with the quote "You need to take good care of your star or it gets all dark and icky." I suspect it was parodying some ad that must've gone like "...good care of your hair..."
A star is just a giant fusion reactor. We just need to add more fusion material (hydrogen) and remove the waste material, which gathers in the core and makes it more difficult for the fusion reaction to continue. It would be possible to prolong the life of the Sun for a while by adding more hydrogen to it, and ideally taking out the heavier elements at the center, which might also be valuable on their own. This seems pretty impossible today, but given we survive enough time until this becomes a problem, then we might find ways to also fix it.
