I loved Olympic National Park! I haven't been to the San Juan Islands, so I can't compare, but I can share a few tips and recommendations for ONP. I don't have kids, but if yours like to be outside and explore, I think they'd enjoy the variety of ONP.
The charm of Olympic is its diversity - you have mountains, rain forests, waterfalls, and beaches - but at the price of being very spread out. There isn't one central location where you can settle down and see it all. However, I will say that I did not find driving between locations to change lodging to be too onerous (I dislike too much time in the car).
I really enjoyed staying at Lake Crescent Lodge and Quileute Oceanside Resort, but I'm a "stay inside the park person", which doesn't give you many options in ONP. You can stay in Port Angeles and reasonably visit Hurricane Ridge, Lake Crescent, and Sol Duc areas, but I think it would be too far for the beaches, or Hoh.
My visit was in May, so while I was able to drive up to Hurricane Ridge, I wasn't able to do any hiking in the area. I think my favorite hike was the Moments in Time/Marymere Falls trail right from the door of my hotel room at Lake Crescent. I'm not sure why it was my favorite - maybe because it was the first day, maybe the weather/lighting was best that day. I also enjoyed Sol Duc Falls, 2nd Beach, Rialto Beach, Ozette Loop, and the Hoh. The Hoh was a little less exciting than I expected because of the other forest hikes I had already done.