I just read this composition in a conversation. Person A tells she doesn't care one way or the other about that, when Person B asks her if she wanted to have an assistant who is punctual or someone who is well-organized. Does this phrase in this context mean, person B wants someone who is either punctual or well-organized?
Yes. It means she doesn't care strongly about either option, so she's equally happy with either. That's if it's a clear choice between punctual OR well-organised.

If she was asked 'Do you want someone who is punctual AND well-organised' and said 'I don't care one way or the other' it would mean she doesn't care whether the person is punctual and well-organised or not. Either option is equal to her.
Thanks for the in detail answer.