"tener un gran dia" means "to have a great day", so you are not wishing someone to have a great day, you are just saying an action. "que tengas" expresses your wish to the other person of having a great day. So the correct way of saying it is "que tengas un buen día" or you could also say "que pases un gran día". I hope I'd help you.

it's like that just because one of the usages of "que" is to say sentences like that. If you say "tener un gran dia", you're saying "to have a good day", but in spanish you have top conjugate the verb "have" according to the person to whom you're talking to. But for one or another reason, "qué" is added before "tengas" for making that phrase.