Unfortunately, the records for Sochaux, Doubs have not yet been digitized but the French are usually very accommodating when it comes to family research. I would suggest you write to the town hall (Mairie) in Sochaux with all the information you have and see if they can be of any help or point you in the right direction. They may have her death record which should give her place of birth and parent's names. You could also ask if they have any marriage records for her children as they can contain quite a bit of information.
The Sochaux 1906 census is online but I didn't find her in it. Later censuses are not online yet.
If you don't speak French, just start your message by apologizing that you don't speak French and they are usually very understanding. They may write back in French and I and others are always happy to translate for you.
One thing to note, in France a woman's vital records are always in her maiden name.
Cité administrative Maurice-Thievent
4 rue de l'Hôtel-de-Ville
25603 Sochaux Cedex
I looked at the 1911, 1921, 1926 and 1931 censuses for Rechesy but didn't find any Stocklins. Do you know what her brother's first name was or whether he was older or younger?