French mathematician Henri Léon Lebesgue's surname is pronounced ləbɛɡ. I am wondering why the letter s is silent. There is a similar-with the definite article-sound word (le) bègue but I cannot find any connection with the surname Lebesgue.
Family names pronunciation is defined by usage. Members of a given family can decide to pronounce their name the way they like among the possible variants.
Moreover, the way family names are written was often unstable in the past, as orthography in general wasn't fixed when family names appeared. When a newborn was declared, their parents were very often illiterate so the person filling the registry wrote what they heard the way they can, not to mention the meaning of the name might have been lost due to the language evolution, or because the name had a French dialect or non French origin.
That means in a single family, children might have slightly different last names than their parents.
In this case and as you suspected, Lebesgue is very understandable and can't be anything but a variant of "Le bègue". Bègue used to be written in various ways in old French, including besgue, beggues, begges, bagues, becgue, baugues, besghes, baggue...
Currently, there are known families named Lebègue, Lebesgue and Lebaigue.
Last names using the form Le + some physical characteristic are quite common in French:
Leblond, Leblanc, Lenoir, Leroux, Legrand, Lepetit, Legros, Legras, Lejeune, Leborgne, Lesourd...