In the Royal Navy there is a commonly used expression called ‘make and mend’ which is nowadays used to indicate a rest break.
It is short for ‘make do and mend’.
It comes from sailship days when ratings were ecused work, ostensibly to maintain their kit and clothing.
Nowadays if you have been on duty in the middle watch 0.00hrs to 0400hrs you normally are given the afternoon off - a make and mend.
There is no entry for make due. make-do (with the hyphen) is a noun meaning “a temporary expedient” or an adjective meaning “characterized by makeshift methods.”
To “make due” would imply that something would be forthcoming, as the librarian impishly pointed out above. It would make more sense to say “come due”, since being due is a pre-arranged condition of lending. And what that has to do with eking out whatever you have to eke escapes me.
To “make do” is one of those strange mental contractions, in which a specific article is implied. As in, “the recipe calls for two teaspoons of vanilla. I only have one, so I’ll have to make [one] do [the job of two].”
In other words, to make do is to say that we have to make whatever is available to us do whatever we need it to do.