It was almost certainly the orphan who killed his adoptive mother.
Why did he do it?
For her assets, for his inheritance.
The question then arises why did it even occur to him to murder the woman who raised him, cared for him, bought him the books he enjoyed studying, and who loved him?
Because he was evil.
It is probable that Guy de Maupassant expected his readers to believe that personality characteristics could be physically inherited, and that since the young man in this story was an orphan he may have come from "bad stock". We know that his birth mother, widowed just before he was born, died penniless, and it was common among the privileged classes who enjoy private incomes to associate poverty with moral degeneracy.
If he is the murderer, he is extremely ungrateful to the woman who raised him, a trait almost universally considered to be deeply unpleasant and sociopathic.
He requests money from her several times and receives it. After obtaining on one occasion a large sum, he asks for more only a few days later. When she says no he appears to be "satisfied with her decision". But we realise he is exceptionally committed to giving a false appearance.
The nasty selfishness of his character is shown in that
he now did not even talk to Madame Source, merely answering her
remarks with short, formal words
He is quiet and terse for a year, and he spooks Madame Source with his behaviour, but nonetheless he is still
agreeable and attentive in his manner toward her
when she expressed a wish, he unmurmuringly carried it into execution.
When she wanted anything brought from the city, he immediately went
there to procure it.
This combination causes her to feel as though "she were going insane".
There is clearly something wrong with him mentally that sadly his adoptive mother's efforts to reach out to him cannot help him get through:
he did not appear to listen to her, and she shuddered (...) when she
has spoken to him five or six times successively without being able to
get a word out of him.
From the viewpoint of the narrator, Maupassant describes him as
a self-contained, unapproachable being, in whom there seemed always to
be some active, dangerous mental labor going on.
After Madame Source's death, he weeps from morning to night, yet Maupassant comments that his state was one of violent grief "at least to all appearance".
How did he do it?
We know that Madame Source's last coach journey was from her new house to her old house. At least this is strongly implied by her body being found by the village postman, whereas her new house is in the suburbs. When the postman notices a splash of blood, he first thinks that
some boozer must have had a nose bleed.
This does not occur to him simply as a possibility: he believes this is what "must" have happened". Why?
The reader is probably meant to infer that the reason for the postman's notion is that the body was found close to a drinking establishment. And the only such place we know about is the cafe at which the orphan spent time that evening. If it was nearby then he would only have had to leave it for a few moments when he heard the coach and then return.
Although the young man appears to have a firm alibi, how firm is it really? If he was in the cafe all the time, then he could not have personally committed the murder and it would be unlikely that the villagers would believe otherwise. There is no suggestion that they believe he paid another party to carry it out or used supernatural means.
If the cafe was a drinking establishment, then many patrons there may get drunk, perhaps even early on in the evening. And the orphan is, or at least he becomes, effective at causing those who believe he is guilty to change their minds about him:
(a) man who speaks with such facility and who is always in good humor
could not have such a crime on his conscience.
Perhaps he has no conscience.
Eventually he even becomes mayor.
Could he simply have manipulated the perceptions of the ten people in the cafe? Perhaps for example they were in two groups and during the crucial few minutes he ensured that each group believed he was in the company of the other?