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Given the fact that you'll only have very limited time in the area, and will have to be back on the ship on time, you might consider primarily using the frequent local trains as it will be a more efficient use of your time. That's also an important consideration to allow time for hiking. The train ride from La Spezia to Riomaggiore (the closest town) is only about 9 minutes.
On the topic of hiking, note that you will require a Park Pass to hike the popular Sentiero Azzurro trails, and only two segments of those are open at the present time. The segment from Monterosso to Vernazza may be a bit "strenuous" for an eight-year-old as there's a steep uphill climb at the Monterosso end.
You could use the Ferry from Monterosso to Vernazza, hike from there to Corniglia and then use the trains from there back to La Spezia on your return to the ship.
Monterosso has the best beaches but you can certainly "jump in the water" in the small enclosed harbour in Vernazza.