Well, grammatically this is absolutely correct. But it sounds very unnatural, we never say it like that.
First: "доброе" literally means "kind". When related to time, it is only used in fixed expressions like "Доброе утро!" - it is a wish, it means you wish someone "a kind morning". In other contexts it does not work. You should use "хорошее" instead, for example: "Сегодня хорошее утро" - "It's a nice morning today".
So, your choice is: "Я надеюсь, что у тебя было хорошее утро?"
Second: However, this expression would only be appropriate if you REALLY want to know if someone's morning was good, it implies you have some reason to doubt that your interlocutor's morning was good. Otherwise, it would be very strange why you ask this question.
The verb "надеюсь" underlines your state of doubtness, as if you were afraid something might have been wrong with your interlocutor this morning.
If you just wish to show your care of someone's life, the best choice is just to ask "Как дела?" or "Как ты?", "У тебя всё хорошо?". The question like "I hope you had a good morning" can be used in very specific situations only.
This is already a cultural point. In Russian culture, when you ask someone "How are you?" (Как дела?), it is understood literally, and the person will tell you a story about his/her recent affairs.
It is not customary to ask everybody "как дела?" in our culture. It is OK to ask your relatives, your friends or someone you know well. Also, a doctor may ask this question to a patient, or a teacher may ask it to a student. It shows you really care about your friend's health/life/business etc. And it is expected to get a honest answer, not just "I'm fine!" all the times as it occurs in the US, for example.
PS Yes, I did not notice one mistake: "было", not "была"!