When someone says "I'll make due" what does it mean?
The idiom is actually “to make do”, and it means to work with what you have, to continue somehow despite an impediment or non-ideal circumstance.
It uses do in the sense of “suffice”, as in “That’ll do”.
A further note regarding the usage of make due (an eggcorn in the question's original wording) and the intended expression make do...
A Google Ngram shows that of the two expressions, make do is currently about 15 times more common than make due, but was rarely used until about 1930. I think it's not too far of a stretch to suggest that the Great Depression may have had something to do with its increased popularity.
Usage of make do seems to have plateaued in the late '30s as economies started to recover, but it surged again from about 1940 (coinciding with World War II), and continued to gain in popularity right up to the 1980s (perhaps coinciding with thrift going out of fashion?). It's yet to be seen whether the recent "re-issuing" of the term following the Global Financial Crisis (see this answer) will transform the plateau from the 1980s into a renewed surge of popularity...
Some supplementary information in addition to Jon Purdy's answer:
In British English (as opposed to American English) due is pronounced like few or queue so it's never confused with do.
To "make do" is a useful expression in tough times such as war or recession. [This British Library page](http://www.bl.uk/learning/timeline/item106365.html] provides an example of both historic and contemporary usage:
Make Do and Mend was a pamphlet issued by the British Ministry of Information in the midst of WWII. It was intended to provide housewives with useful tips on how to be both frugal and stylish in times of harsh rationing. With its thrifty design ideas and advice on reusing old clothing, the pamphlet was an indispensable guide for households. Readers were advised to create pretty ‘decorative patches’ to cover holes in warn garments; unpick old jumpers to re-knit chic alternatives; turn men’s clothes into women’s; as well as darn, alter and protect against the ‘moth menace’. An updated version of the book was recently released to coincide with the economic recession, offering similar frugal advice for 21st century families.
I always think that "make due" would make almost more sense (I realize the idiom is actually "make do"). "Making due" could imply accomplishing something owed/expected despite obstacles, which is really the sense of the phrase.
make due means the same as make do, to get along with little, to reach a certain goal.
It is often called a spelling mistake, but there's no established dividing line between accepted and unacceptable spellings. It's however not a subjective issue. If somebody, objectively, does not think of the verb to do in this idiom, than it's not a spelling mistake. The issue is more complicated.
It's perfectly reasonable to assume this stem from an expression to make dues to get by or from to make due restitution, but that should be hard to prove. Americans, barring knowledge of French and English history, have a hard time fathoming this. And due to lack of historical sources, it should be hard to prove either way.
to make ends meet has a ring of direction towards a goal. To pay the bills and save money for food, debtors may scrape together all their pennies, to make do.
Also note that lend or borrow do not denote the same directions of lending everywhere. Colloquial speech is prone to corrupting legal terms. Hence somebody who has to make due, will later have to pay due. The -s there might be a possessive clitic 's, reinterpreted as plural. I don't know.