Best way to stop a bathroom exhaust fan from rattling?

Rattling exhaust fans almost always come down to one of a few causes, and the fix is usually quick.
Start by cutting power at the breaker, then pull off the grille cover (most just pull down and you squeeze the spring clips). The most common culprit is simply dust and grime buildup on the fan blades, which throws the wheel off balance. Vacuum everything out and wipe the blades clean — this alone fixes a huge share of rattles.
If it still rattles, check these in order:
Loose grille or housing screws. Vibration works them loose over time. Tighten the screws holding the housing to the joist and make sure the grille clips seat firmly. A thin foam weatherstrip tape around the grille edge where it meets the ceiling quiets a buzzing cover nicely.
The fan blade wheel itself. Spin it by hand — it should turn smoothly and freely. If it wobbles, it may have slipped on the motor shaft; many have a small set screw you can re-tighten. If it scrapes the housing, gently recenter it.
Worn motor bearings. If the noise is more of a grinding hum and the wheel feels stiff or gritty when spun, the motor is on its way out. Most fans have a replaceable motor/blower assembly that plugs in and swaps out in 15 minutes — cheaper than replacing the whole unit, and you don't have to touch the ductwork.
The damper flap. Sometimes the rattle isn't the fan at all but the backdraft damper in the duct clacking in the breeze, especially on windy days. You can dampen it with a small piece of adhesive foam where it lands.
If the fan is old and loud even when clean, replacing it with a modern quiet model (look for something rated around 1.0 sone or less) is often worth it — new units are dramatically quieter and usually fit the existing housing hole.

Most rattling comes from dust buildup or a loose fan blade/motor assembly, so cut power at the breaker, remove the cover, and clean the blades and housing thoroughly. Check that the mounting screws, fan blade, and motor bracket are all tight, and make sure the blade isn't scraping the housing—you can bend it back slightly if it is. If the noise persists, the motor bearings are likely worn, and replacing the motor or the whole unit (often $20–50 for a motor) is the fix.