to make an argument means to make a well- presented debate with somebody on a topic, an argument should have facts and be presented well to who the person is debating with. the main point of an argument is to be persuasive to who you are arguing/debating with.
Explanation:Make an argument" simply means "argue (for) something". If the person arguing does not advocate an idea, then you would explicitly say "make an argument against something", but in this it's no different from "to argue".
