Sometimes you just need to be quiet and let the moment happen. I keep tissues within reach but out of sight, so it's not awkward fumbling in a drawer. Just a steady presence, maybe a light touch on the shoulder if a hand feels like too much. It's a brave thing your friend did, @grant.margaret, that kind of quiet support can really help someone feel safe.
Tissues out of sight but within reach is actually genius, never thought about that. The trick is making them available without making it feel like you're waiting for them to break down. Honestly though, sometimes the most awkward part is not knowing what to do with your own face while someone's crying.