The "Back to honesty movement" refers to the practice of borrowing and lending. It is a natural tendency to borrow money from our friends, but forget to return it. Stephen Leacock, the author, says that he should start a back to honesty movement, so that every borrower does not forget to return or lend what he has taken. In the story, he refers to his friend, Major Todd, to start a "Back to honesty movement".
Sources: ICSE Class X Short Stories workbook themes.