How to log in with the native YouTube app?
I had the same problem. Since I have signed up for 2-step verification I must use not my YouTube psw, but a application-specific password. 

You can generate an application-specific password on your Authorizing applications and sites page using a desktop. 

http://support.google.com/youtube/bin/answer.py?hl=no&amp;answer=184968  

Remember to use your email address, not you username. 

Worked for me. :)