When someone falls down and you give your hand to her but she doesn't hold your hand, what's the proper word for the gesture you've done. You extend your hand? Is "extending a hand" right usage as a verb? Is there a better saying for the movement that you make when you are giving your hand to someone?
"Extending your hand" is indeed correct, understood exactly in this way, and of long use for precisely this. At least one dictionary even lists "extend your hand" as a phrase with precisely that meaning. The OED doesn't, but does have a definition:
To stretch forth (the arm or hand). Also, to hold out, put forward (a staff, etc.).
And includes examples of Shakespeare and Dryden using it in this way.
All in all, you couldn't ask for a more apt word choice.
Both extend and offer are correct and current usage.
"As we were walking I offered her my hand."
"I offered my hand and he shook it."
"I approached him and extended my hand."
"He stared at me for a moment and then reluctantly extended his hand."
Reach out is a useful expression:
Lit. to extend one's grasp outward.
- He reached out, but there was no one to take hold of. I reached out and grabbed onto the first thing I could get hold of.