Whale watching is not the fun you might expect!
It is the single most cold thing you can do! (may be exaggerating but do take my point about needing proper warm clothing suitable for the wet conditions and knowing how to layer your clothes - don't buy cheap gear and think it will do - it wont).
Its actually not something I put at the top of my list to do and its seasonal anyway so definitely check the time of year you are going.
Generally speaking, you do need to be aware of what time of year you are going because so much is seasonal in Iceland.
The Golden Circle is ok. But its utterly rammed with tourists these days. You are better exploring it yourself and making sure you get up early and leave Reykjavik before the coaches do! (I'm serious).
But I do think there are better things to do than simply the beaten track for tourists tbh.
Forget the Blue Lagoon too for the same reasons. It was excellent 20 years ago. There are a couple of other natural hot springs which are better now. I think the Hidden or Secret Lagoon (on the gold circle route). I also rate the swimming pools in the city as good fun.
Jökulsárlón was a highlight for me, but its a fair trek from Reykjavik. You can do it in a day but its a long drive. Its better if you can find an overnight stop nearby (easier said than done!) to make the most of it.
The waterfalls along the southern ring road are amazing for all ages.
What I would recommend is doing an organised trip for something a bit out of the ordinary. There are so many different ones available but depend on the time of year you go.
You can go into a glacier (with off road coach to get there), lava tunnel tours where you go in a volcano, kayaking tours, climbing tours, glacier walk tours, tours to the interior (again involving off road transport), snorkeling tours, horse riding tours... They can be pricey but honestly worth the money and 100 times better than the standard golden circle tours (which are better in your own time and on your own power). Try and pick at least one as your main event for your trip if you can (our budget has never managed more as eating out is pricey). The vast majority will cater for kids the right age for you.
It depends on whether your kids might get bored if you are thinking about aurora tours. Not sure all kids will appreciate and be patient enough for it.
In terms of saving money, rent an apartment and self cater as much as you can. Iceland is great for eating out but expensive - do your research on where you want to eat out and book in advance (can't stress this enough especially for the best places). Make sure you eat lamb or fish out (or both!) at least once.
Its a place you go to walk and be outdoors more than anything. If your kids are good with this, they will have a blast. If they would rather do anything but go for a walk it might be a lot more challenging as you can't escape it really.