I haven't done the timeshare, but have been to the Kona side of the Big Island multiple times. I have family who lived there for many years, and my family has always loved visiting Hawaii (which is why some relatives moved there for a time.)
As others have said, rent a car, preferably 4 wheel drive. Make sure you go to the black sand beach. There aren't many in the world, so it is a moderately unique experience. Go to Volcanoes National Park. It's a drive, but definitely worth it, again, because it is a fairly unique experience. You can continue the drive to the Hilo said for more traditional, tropical feeling beauty. (Kona side is beautiful in it's own way, but parts of it feel very stark because of the all the visible volcanic rock.) There are nice botanic gardens and waterfalls, plus a huge banyan tree, on Hilo side that really make you feel like you're in a lush, tropical paradise. Particularly if you haven't been to Hawaii before, and you're expecting beaches and tropics and greenery, you need to spend some of your time at least one day around Hilo.
If you're the adventurous type, and you don't mind violating your rental car agreement (which will almost certainly say don't take the Saddleback road), go up to the observatory tourist center at night (you can drive there during the day, but plan to stay until after dark). (Pack a warm jacket and have warm clothes with you if you decide to do this.) The views of the stars, even from the visitor's center where local amateur astronomers set up their telescopes, are like nothing you've ever seen before. Seriously, this is an amazing experience. The very long drive back to Kona side in the dark on a not great road (the aforementioned Saddleback) is an experience of another sort, and one I would gladly forgo... but you can't do the observatory at night without doing the drive. It _is_ stressful; it is bumpy because it isn't well-maintained and dark with no places to stop because a lot of the road goes through land owned by the military. Make sure you have a full tank of gas. Only you can decide if it is worth it for your family.
The Big Island is big, but if you're willing to do a very long day, you can manage everything I've talked about in one day. We've done it when doing the hard tourist sightseeing thing - start nearish Kona, drive to the black sand beach and walk around and point at sea turtles, drive through volcanoes national park (stopping at several key spots to explore), drive to Hilo and see a few sites, drive to the observatory visitor center and look at stars, then drive back to where we were staying in Kona. Now, I think it ended up being about a 17 hour day because of the late night at the observatory, but it was doable for a group of adults. I would take it in smaller pieces with kids.