The big problem with ONP in the high summer is a general scarcity of accommodations, especially on the coastal strip. There are accommodations around La Push, at Kalaloch, and around Lake Quinalt, but they fill up fast.
I'd focus on two areas. First, the area around Lake Crescent (near Port Angeles) has some good hikes and walks (Hurricane Ridge as previously mentioned) and Lake Crescent itself is gorgeous.
Second, out on the coast, the beaches near La Push and Kalaloch are superb (try Rialto Beach, Ruby Beach and others) but IMO the crown jewel for walkers and explorers is the Hoh Valley rain forest. The beach scenery is stunning; the Hoh valley is like another planet.
Specific timing is important; there are big differences from early summer to late summer, particularly on the coast. When were you thinking?