Is there any recorded instance of a passenger's hat or scarf being blown into an aircraft's engine?
If you are boarding with an engine running, it's going to be the one on the opposite side running with the one on the boarding side shut down.  However, there's little doubt that items like that have flown off heads in high winds and drifted across to the opposite engine and been ingested, in all the millions and millions of flight hours over the decades.

In any case, something light enough to get carried off by the wind like that would just pass through the fan, or, if it was sucked into the core, get chopped to little pieces, burned and come out the back as ash particles.  A large enough item would probably cause enough flow disruption to affect the engine's idle and give a turbine temperature spike, but it's unlikely to do significant damage unless it was, say, an entire coat or something, and that would required a mighty windy day.

What's biggest Foreign Object Damage threat to engines on the ramp?  Luggage zipper tags.