GMMichael
Guide of Modos
For example my sister's character has a trait that she is often fiddling with things and dropping them by accident.
She picked up an axe that was on display in a weapons shop and dropped it.
Is this something that deserves Inspiration or not?
Short answer: yes. Everything deserves inspiration! The main reason is that characters get only one inspiration at a time, so why the heck not? I believe they can award it to their comrades as well, which means that if your PC has inspiration, he can save you the trouble and remember to award it to his buddy.
For the record, it's pretty cool that your PC goes around dropping stuff (and remembering to roleplay it) all the time.
You have to be more careful when you start to realize that PCs are getting and using inspiration on just about every check they make. Only then do you say, "okay, maybe I'm being to liberal with the inspiration." That's when you use this criterion:
Did the PC just do something harmful to his character or character goals, yet is inline with what his character would do?