"Její jméno já neznám."

Translation: I do not know her name.

[blublevita]

Does the "já" here serve the purpose of emphasis?

[ValaCZE]

Yes it is because of emphasize "já".

[unci14]

Would the word order "Já neznám její jméno" also be possible?

[BossKaiden]

I think that if you did that you would basically be saying, "I know her, but not her name." Since you're putting emphasis on "jméno".

[Janmunroe]

Would Jeji jmeno be acc. sing. neuter here? My brain is clogging up at the moment.

[VladaFu]

yes

[Vlad_Lesnievski]

As I understand
"I do not know its name" and "I do not know their name" would be legitimate translations as well, wouldn't they?

[VladaFu]

Certainly not their (jejich). Perhaps "its" if we consider a not-living entity that is feminine in Czech. But it certainly is feminine here.