Yes, '你早' is a perfectly good way of saying good morning. Although, I would be more likely to use either '早安' or more simply ‘早’. I have heard some students of Chinese say that they don't like using ‘早’, because of it's brevity and the fact that it feels discourteous. However, don't worry about this. It is more likely to be used between friends, that is true but it is also a trait of Chinese to be brief. Keeping it short and sweet is what we like.
Having said this, use what you feel comfortable with. These options are all appropriate and people will understand and respond all the same.
Keep up the good work!
- Lin Ping