... ...
If it could be nudged accurately enough, it would be worth putting it into orbit around the Earth or Moon for potential future resource exploitation.
Step one is we learn enough to nudge a killer asteroid to miss us so we get to play another day. Then we learn how to manipulate those object's orbits to make them "near by" resource depots.
After a few millions of years of technology development, we should be able to move much large object and use those large object's gravity to alter Earth orbit. Around Pluto, huge asteroids are aplenty. So a
spinlaunch here, a sling launch there, and a rail launch here and there, we got a collection of large objects in solar orbits of our own design. As the Sun get bigger and bigger, our own Earth orbit expands because of the gravity of the objects we put there to shape our orbit by design... Now we get to watch the red Red Giant at a safe distance. We get to continue playing our game instead of roasting in the Red Giant Sun's atmosphere.
So, now that we survived the red giant phase, next thing we worry about is how do we shield ourselves from the Sun going NOVA...
At about the 1bn year mark, Andromeda Galaxy would be joining ours. That may put our solar system adrift - even before our Sun goes red giant. We will have to play that one by ear. Let just say we go with the probability that our solar system will stay intact.
If we shielded ourselves from the danger of our Sun going nova, we are still in the game then. We need to find some way to get resources from outside our solar system. That white-dwarf that was our Sun isn't going to give us much. But we will suck up as much as we can from our Gas Giants. That will keep us in the game a while longer. But sooner or later, we live or die by if we can unchain ourselves from our solar system.
Note: Bolded text highlights the fact that I have to stray away from main topic - spinlaunch.