Some French names come in slightly different variant spellings, and the different spellings are more prevalent in different regions of French Europe:
- Michaud (Jura & Charente Maritime)
- Michaut (Yonne)
- Michaux (Nord, Belgium)
- Micheaux (Nord)
- Micheau (Charente Maritime & Allier)
- Michau (Eure et Loir)
Similar variants exist for other names ending in the sound /o/. How did these variants develop?
There’s no standard spelling as far as proper names are concerned. Proper names were used orally well before the majority of the population could read and write and when they started to be taken down by clerks for official records (for the État Civil in France in the 16th century) original families with that name had already dispersed all over the country and the colonies and the clerks who wrote them down all had their own way of doing it, just writing down what they heard.
All variants of the name Michaud, Michaut, Michaux, Micheaux, Micheau, Michau, Michot, Micho, etc… are pronounced in French in exactly the same way : [miʃo], whatever consonant you add at the end will not be sounded and the spellings eau, au and o are pronouced /o/.
That’s probably the case not only for French surnames but for a lot of languages and countries. But we can expect to find a great number of people spelling their names in the same manner within a restricted area since for a very long time migrations were slow and people did not go very far from their original area.
I’ve edited the answer to point out to the Namespedia database. It will enable you to see the distribution of the name in its different spellings across the world, including the different regions of France or other countries where the name has spread.