"لا أَعْرِف اِسْمها."

Translation: I do not know her name.

[T-MAN9000]

What would be 'I do not know his name'?

[TJ_Q8]

I don't know his name = لا أعْرِفُ اسمَه (lá a3irfu ismah).

[StevenWest12]

So why does it sound like "اِسمُها" while it is written as "اِسْمها"? Is there a grammatical relevance for the "u" or is it purely phonological?

[TJ_Q8]

Sorry, put on an answer first and realized I was mistaken with something.
The audio is correct. It is اسْمَها because the word اسم (name) is in accusative position, which requires (-a) on the last letter. Thus it is (Ismahá).
Apparently though they didn't type the FatHa on the (M) in the Arabic phrase above. The little circle on the (S) is Sukún, which notes that this letter is stable (i.e. not moved by any vowel).

[Ibraahiim13]

How to tell which harakat to use?

[TJ_Q8]

On words endings?

• (-u) is for nominative (normal) noun.
• (-a) is for the accusative case.
• (-i) is for the prepositional and genitive case.

This is concerning the nouns, in simple format. Things change with plurals and duals, as well as some special categories of nouns. Anyway, the above list is the basic one.
In addition, all of the above endings change to Tanwin ending when the noun is indefinite (i.e. they become -un, -an, -in respectively).