Use of apostrophe

Grammatically ok. But I think this sounds better:
"We will never fully succeed in achieving Mr. Obama’s vision of a poor girl having exactly the same opportunities as a wealthy girl."

I think so Agm, abbr of "girl as"

Probably means:

. . . of a poor girl’s (desire of ) having exactly the same opportunities as a wealthy girl."

Dahileman and Punkey, both wide of the mark. In this piece...

"a poor girl’s having exactly the same opportunities as a wealthy girl"

The apostrophe in girl's is the possessive apostrophe. Obama is being quite formal and is using the possessive before a gerund.

John's winning the race made me happy.
Mary objected to my borrowing her dress.
Your leaving early was a wise decision.
Bill's having a driving licence is an advantage.

In informal writing, there is a trend toward dropping the possessive before a gerund. We often use a simple noun or an object pronoun instead:

We celebrated John winning the race.
Mary objected to me borrowing her dress.
You leaving early was a wise decision.
Bill having a driving licence is an advantage.

However, in formal writing, the use of the possessive form before a gerund is still preferred. Also, the possessive form may be important for clarity. Consider the difference between the two examples below:

Bill is in favour of the candidate being interviewed on Friday.
(Bill prefers the candidate who has an interview on Friday)

Bill is in favour of the candidate's being interviewed on Friday.
(Bill wants the interview to be on Friday)

Yea Con, now that you mention it….

Contrex 1, Rest of the World 0

Wot Tag sed.