Afer the Battle of Tours, Charles Martel was nicknamed "the Hammer". In modern-day French, "marteau" means hammer, so I was wondering what came first? Was Charles called the Hammer because of his smashing victory AND because his last name sounded like the word "marteau", or did the word originate from his last name?
I think you misunderstood his name, "the hammer" is the litteral translation of "Martel". As you noticed, "martel" sounds a lot like the modern-day word "marteau", which is explained by the fact that "martel" is the ancient way of saying "marteau" (just like "hamor" is the Old English way of saying hammer). "Martel" even appears in contemporary-French, for example in the verb "marteler" (to hit with a hammer).
Besides, "Martel" was not a family name, but a nickname. Frankish people usually didn't have a family name. They, could earn nicknames though, "Martel", "The bald", "the pious", etc. depending on what they did during their lives.
So to answer to your question, he earned his nickname "Martel" because of his smashing victory, that made people say he was like a hammer (martel).