Neat place, but it's not in Santa Cruz
Lick Observatory is definitely a destination very different from traditional parks and museums. Lick Observatory is one of the oldest astronomical observation points in the United States, and its atmosphere really reminds you of that.
When visiting, you don't get to look through any telescopes (that's can be done via signing up). You however do get a volunteer to tell a story about how the observatory came to existence, and the personalities involved -- a very interesting story indeed.
It's really not a great place for you children since there isn't any sort of interactive exhibits, but it is a nice place that gives you the feeling that you're on top of the world (on a nice day that is). There is a small gift shop with everything Lick Observatory, or astronomy that's actually reasonably priced. You'll probably end up with a one-of-a-kind shirt to share.
From the Evergreen area, it takes about 30 minutes to traverse very narrow and windy roads to the Mount Hamilton's summit. Start with a full tank of gas, and bring food -- none is available between Evergreen and Mount Hamilton. If you really want to, pack picnic -- Joseph D. Grant Park is along the way and has a nice picnic area.
There is no charge to visit Lick Observatory, but a parking fee may be charged at Joseph D. Grant County Park.
BTW -- It's connection to Santa Cruz rests in that the observatory is run by the University of California, Santa Cruz.