Can you say this?
"She gave him the book out of his persistent asking."
The online dictionary says "out of" means: because of, owing to...
If you replace "out of" with either of those phrases the sentence is acceptable, so shouldn't the former be also? And, yet, something seems awkward about it. :)
I agree that it sounds strange and I think (but would like others to confirm) that it is because you cannot follow out of with a gerund - asking in this case. You could use out of followed by, e.g., spite, duty, bravado, etc.