This is a case of gerund-participial non-finite clause. CAGEL explains that:
In formal style the subject of a gerund-participial clause that is functioning as complement (of a verb or preposition) appears in genitive case, as in
- She didn't approve of [his being given a second chance]. (p. 110)
EDIT: Thanks to the sharp eye of some users, I realised the confusion between her genitive and her accusative. CGEL helps us here too. On page 210 it gives this example:
She insisted on [my / me being present throughout the interview]. [here you could replace the pair my/me with his/him]
And it explains that:
The clause between the brackets is complement of the preposition on. If a personal pronoun subject is not in the genitive, then it takes accusative case... The choice between genitive and non-genitive depends partly on style and partly on the type of NP. The genitive is characteristic of fairly formal style, and overall it is relatively infrequent. It is most likely with personal pronouns, and next most likely with short singular NPs denoting humans.
The book goes on to enumerate cases in which genitive subjects are not possible, like in this example:
She sought advice from Ed, [he being the most experienced of her colleagues]
No genitive possible here, and the informal accusative him would sound out of place in this formal sentence.
So there you go, in you particular case
She didn't approve of him being given a second chance.
is perfectly grammatical but less formal than
She didn't approve of his being given a second chance.