Just remember it's the same as "por la noche" for "at night, in the night". Monday night = El lunes por la noche. Monday afternoon = El lunes por la tarde. Monday morning = El lunes por la mañana.
"Mañana" just happens to mean "tomorrow" as well as "morning", so if you want to say "tomorrow morning".... "mañana por la mañana". It looks odd, but it's really the same phrase you'd use to refer to any other morning!
I think it depends on the context. I guess in most conversations they'd leave the 'mañana' meaning 'morning' away but I could imagine then saying it if it were something like this: 'Tomorrow in the morning I'll call you, on Friday morning I'll visit you and on Saturday morning I'll give you a present'.