Maitre Hauchecorne, a miserly and petty man, bends down and picks up the thin piece of yarn because it is free, and he thinks it may be useful at some time.
The key to understanding the character of Maitre Hauchcorne is the phrase that Guy de Maupassant employs in the sixth paragraph:
...Maitre Hauchecorne, like the true Norman he was, thought it worthwhile to pick up anything that could be useful; and he stooped down painfully, for he suffered from rheumatism.
In this tale of pettiness, the author Guy de Maupassant, a Norman himself [from the former province in northwestern France, where the beaches were stormed by US troops in World War II], writes a withering social criticism of the Norman peasants, whose suspicion and envy of one another appalled him.
In Maupassant's psychological examination and naturalistic and depiction of these Norman peasants, Hauchecorne's initial deception about not having picked up the string and his petty nature which probably would have led him to take the lost purse given the opportunity, work against him as his protestations to the authorities are excessive and fuel the suspicious nature of the others.
All this behavior underscores the attitude of the author Guy de Maupassant, who worked as a government clerk in Normandy and once declared of the Normans, "Everyone is perfidious, a liar, and a phony. Everyone wears a false face."
I think the simple answer is Hauchecorne picks up the string because he can't miss any opportunity to get something for nothing. Maupassant uses the word "economical" -- "Maitre Hauchecorne, economical as are all true Normans, reflected that everything was worth picking up which could be of any use" -- when describing how Hauchecorne, even though racked with rheumatism, bends over to pick the string up. Another way of saying it is that he is "cheap" or "grasping" or "greedy" -- there is a sense that, for Hauchecorne, this is more than principle; it is a compulsion. It seems like a small thing, but this moment contains a deep truth about his character that Malandain, who watches him pick up the string, recognizes instantly. Malandain is his enemy. It comes as no surprise to the reader (or, probably, to Hauchecorne) that Malandain is his accuser when Hauchecorne is questioned about the missing wallet. Both men know that, had Hauchecorne had the luck to find the wallet instead of the string, he would have no doubt taken it.