Does the "já" here serve the purpose of emphasis?
Yes it is because of emphasize "já".
Would the word order "Já neznám její jméno" also be possible?
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".
Would Jeji jmeno be acc. sing. neuter here? My brain is clogging up at the moment.
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?
Certainly not their (jejich). Perhaps "its" if we consider a not-living entity that is feminine in Czech. But it certainly is feminine here.