According to the Oxford English Dictionary, the "s" of "bridesmaid" found its way into the word during the 19th century. The earlier form was "bridemaid."
Many centuries ago, the noun "bride" referred to the formal joining together of a couple "” the ceremony itself "” but over the years it came to refer to the female partner of the couple. The "s" was added in the 19th century in the mistaken belief that it should be a genitive ("possessive") form. It was thought to mean "bride's maid."
Other compound nouns that started out without the "s" and then acquired it in the same way are
groomsman = "man who attends the bridegroom"
bondsman/ -woman = "man/woman of bond" (indentured servant)
kinsman/ -woman = "blood relative"
gamesman = "one who is good at manipulating people and events for one's own advantage"
There must be many other examples, but I can't think of any more to look up ("ombudsman" from Swedish, didn't develop the same way). Maybe some of our other members have suggestions.