Depends on the mission criteria - but weather definitely matters! Our teams always have a GO / NO-GO decision point the day before the scheduled launch. We'll discuss all the implications of predicted weather for the next day and decide whether to pull the trigger or not. Once we decide to launch the day before, we almost certainly follow through.
Clouds won't necessarily break a flight. Weather monitoring balloons may actually see this as a plus (for weather variety). Clouds can obscure cameras as you pass through them, but often the views can still be spectacular once you're above the cloud layer. Clouds may block ground features if that's what you're looking for.
Precipitation is usually a no-go for us. I've never launched in the rain. The main concern is that it interferes with balloon ascent, in addition to set-up, launch and recovery. It can also be uncomfortable as a spectator. Clouds also come with precipitation so there are the same issues as above.
Lightning / thunder on their own are not necessarily deal breakers. I can imagine some pretty slick camera views with lightning in the shot. However, I would not recommend unless you are an experienced balloonist. Since you'll likely be launching from a large open field, you're much more exposed to lightning strikes during ground operations, especially with the tall balloon.
Depending on the terrain, winds are a bigger concern than clouds. When we launched balloons near the coasts, there was always a risk of overshooting into the ocean so we were always careful with wind margins (and sometimes overfilled our balloon just to make sure it would pop before reaching the water). Even if you're some distance from water, higher winds usually means a longer chase and a larger landing dispersion.
For beginning teams, I'd say clear skies and low winds are the target. But I wouldn't delay too long for "the perfect day" because you never know for sure how weather will actually unfold during the flight.