As with so many other English idioms, the confusion arises here simply because of the way the words sound. It’s the same reason why people may mistakenly talk about nipping it in the butt or extracting revenge. The fit common words into these sayings, because they think they make sense.
In this particular case, the idiomatic phrase that you want is “make do.”
I really prefer Coca-Cola, but I guess I’ll have to make do with Pepsi instead.
The more desirable cola for this person is unavailable (for some unstated reason), so he has decided to drink Pepsi in place of Coke. There are a few other related definitions for “make do.” If you’re thrown into an unexpected situation where you need to improvise or “wing it,” you might say that you’ll “make do.” The “do” refers to serving a particular purpose.
I don’t have a hammer, but the bottom of this screwdriver should do (the trick).
Since there’s no hammer, I’ll make do with the bottom of this screwdriver.
Part of the reason why some people may (incorrectly) write the phrase as “make due” instead is that it almost makes logical sense. The word “due” can mean appropriate, sufficient or reasonable, as in the phrases “in due time” and “due diligence.”
In this way, saying that you’ll “make due” with some alternative (like Pepsi and the bottom of a screwdriver) could be interpreted as meaning that the alternatives are reasonable and sufficient substitutes (for Coke and the hammer). You could argue the legitimacy of this usage, but it is still ultimately incorrect. “Make do” is the term you want.