Ok: A few reasons why Osgood would make a perfect companion for the Doctor:
1) She has absolute faith in the Doctor. She has likely never met him before, but she is absolutely sure on multiple occasions that he will make everything better. She doesn’t question or wonder, she truly believes that the Doctor will save her.2) She’s damned clever. She figures out within -seconds- of reporting the analysis of the stone dust why the statues were destroyed. Not only that but then she tries to save her colleague by -calmly- warning him, inspite of being obviously terrified, that they are in danger.
3) She was threatened, attacked, and generally abused by the Zygons, and yet still, when the Doctor’s memory loss trick was wearing off, she not only helped her Zygon twin whilst they were experiencing an asthma attack, but also kept the fact that they had established who was who a secret, because she knew how important the treaty being established around them was. Even after everything they put her through, she understood the Doctor’s plan, how important the treaty was, and displayed instant compassion at the sight of her attacker in distress.
4) Even though she’s painted as being ‘not conventionally pretty’ (rolls eyes) and is generally a stand up stereotypical ‘geek’, she kicks butt. Look at her with that scarf!
The Doctor needs someone like her. Someone he can’t talk down to, someone intelligent and sensitive, extremely brave, who can remind him not all human girls want to get into his boxer shorts, some of them just want to be good.