The second example is correct and the first example is not correct. "Of" is a preposition. "Off" is also a preposition, but it is meaningless here. "Of my hand" tells what she is not supposed to let go of. (Keep holding my hand).

Of means : from where. Centuries ago... so it is said... of was used to tell where a person was from - and it was used as part of their title or name. I would be Jesse of America, and if you went outside of your country, you would be Foralltime of Viet Nam. I do not believe that 'of' was pronounced the same back then, or even spelled the same. I do not know. But today, this word 'of' has the meaning (form where) that the word we are discussing also had. It is used much like the word which. They are words that say "specifically this_______" and the speaker fills in whatever they want to refer to. Don't let go.. What shall I not let go? specifically my hand.

Don't let go OF my hand. Off means something different and would never fit into this sentence,

"Of" is correct in your example. "To let go" means "to release" (figuratively, "to abandon" or "to put aside"), and it is usually followed by the preposition "of." Whatever was released becomes the object of the preposition, in this case "hand." Other examples: "The monkey let go of the top branch of the tree and fell to another branch below." "Don't let go of your dreams." "My sister let go of the balloon and watched it fly away." "His doctor told him that he must let go of the past."

"Off" is correct... For example: Face "off" , Windows "of" House...