Dock lines (and anchor rodes) should be stretchy. Thicker is not necessarily better.
We have some 3/8 and some 1/2 dock lines. All double-braid nylon - nylon for stretch, and double-braid because it doesn't tangle as easily as 3-strand, and my crew much prefers it.
We've settled on the 3/8 dock lines for normal usage - as others have said, they fit more easily in the cleats - even in my oversized 8" bow and stern cleats, and definitely in the standard 6" ones (which I reused as midship cleats). We carry a few 1/2" dock lines as backups, and we'd use them in really rough conditions. But most of the places we dock are well sheltered; if we moored at a marina that was exposed to chop, I'd probably want to use the 1/2", but I wouldn't want to leave the boat moored where she wasn't well sheltered anyway.
Our anchor rode is 1/2" NER 3-strand. If I ever replace it, it will be with 7/16" Novabraid. Nearly the same strength as our current 1/2", similar stretch (energy absorbtion), and easier to store in the bow (smaller diameter, and double-braid stacks tighter than 3-strand). The storage space matters in the Puget Sound, where the tidal range means we need to carry 250'+ of rode. Maybe not so much if you only need 100' or so.