It is fairly obvious that the names Dostoevsky uses for the characters in The Brothers Karamazov are important. Names are not only labels for identifying characters, but the meaning of names also point to their type of character or role in the overall theme of the story. For example, the name of the hero Alexei is the Russian form of Alex (from Greek), meaning helper or defender of mankind.
But I wondered what does the name Karamazov mean? The answer comes in Book 4, Chapter 6, “The Strain in the Cottage”, when Arina Petrovna, the mother of the boy Ilyusha, addresses Alexei Karamazov as “Mr. Chernomazov”. The footnote to my edition adds,
Chernomazov: Arina Petrovna inadvertently brings out the implicit meaning of Alyosha’s surname: cherny is Russian for “black”; however, in Turkish and Tartar languages, kara also means “black” (the root, maz, in Russian conveys the idea of “paint” or “smear”).
So, Karamazov means black smear, as in sin, or the stain of original sin. Seems fitting from what I know so far of most of the Karamazovs. The black smear seems to rub off onto nearly everyone who is not vigilant.