Exercise 7.9 in this book:
Given that any set has a multiple-choice function, ie a function picking out a nonempty finite subset of each nonempty subset, show that any totally orderable set can be well-ordered.
To get a well-order we need to modify the total order so that each subset has a least element. For each subset we have a finite subset available, so we can just change the order so that all elements from that finite subset are put below the remaining elements? Does that work?
A non-empty finite subset of a totally ordered set has a least element. Thus, a multiple-choice function for non-empty subsets of a totally ordered set induces an ordinary choice function, and you can then use the usual argument to get a well-ordering of the set.