It is for sure a star in "the" making. I have never heard anyone use a star in making before and it sounds rather off to me as well. This phrase always uses the article "the". The reason is that "the" is used before Present Participle and before adjectives and verbs functioning as nouns. For example The singing, The rolling, The delaying, The sighting etc.. All of these are verbs with ing but they can be functioning as nouns depending on the context they're in.

A star in the making is correct. It's kind of like 'a star in THE process of being made'. Easiest just to consider it a saying.