The villain dies, his daughter shows up, walks up to the hero... and thanks him for killing the villain, whom she thought was a terrible person.
Alternatively, the villain dies, and his brother shows up, thirsty for the hero's blood. The hero starts with the usual speech that "revenge won't bring your brother back", but the brother laughs it off. Turns out he's not trying to avenge the villain, he's just pissed that the hero was the one who killed the villain and nothim.
After the villain's death, a relative of his shows up but appears to show no desire for revenge. Then another relative appears, and he's not of the same opinion there.
Alternatively the relative/friend/loved one of the villain seems to be OK with this but it is then revealed that it was all an act and was plotting revenge but was only being nice about it at first just to try to get the heroes into a false sense of security.
Parodied: The villain's son shows up, thirsty for revenge... and six years old. He attempts to fight the heroes, who fail to take him seriously, much to their regret.
Zig Zagged: The villain is dead, and his daughter shows up, but she wants to join the heroes. Great! Except it was an act to get closer to the heroes and then them. After a fight, the heroes show her evidence that the villain was pure evil, and she seems to back down... but it was just to get the heroes' guards down. After a while, she realizes that revenge won't bring her father back, and decides to do something constructive with her life. And then the ghost of the villain possesses her and tries to kill the heroes.
Averted: The villain is dead. Nobody comes in to avenge him.
Enforced: "We can't just end the series now — we could still milk more money out of this! Let's have someone who knew the villain come in to take his place."
Lampshaded: "He had friends? Uh-oh, now they're mad at us."
Invoked: The villain knows he'll probably be defeated soon, and sets out to make himself some friends to take over his fighting the hero if the situation comes.
Exploited: The villain's brother waits for the villain to be killed so that he can finish off the heroes and pass it off as this trope. Thus eliminating both obstacles to his Evil Plan to Take Over the World.
Alternatively, the heroes make sure to take the villain alive so that his friend and family won't see them as murderers.
Discussed: "It's like some kind of book — kill one, another one pops up to take his place, repeat."
Conversed: "Oh, come on! Just end the series — it won't be the same with the villain's mother as Big Bad — talk about milking the cash cow."
Deconstructed: The dead villain's friend shows up and reveals that the villain was truthfully not as bad as he looked- instead of the monster the heroes thought they were killing, he was more of a Well-Intentioned Extremist, with friends and family.
The villain may not have been so bad, or was at least truly loved by his avenger, but vengeance eventually corrupts the new antagonist.
The new "villain" is fully aware of what kind of person they're avenging, but is bound by honour to avenge them anyway. They eventually end their conflict with the heroes by simply having them repay the debt through service to the villain's kingdom (which is pretty nice with the new guy in charge).
Played For Laughs: The villain was the head of a large, branching family, and in an inspired moment of Genre Savvy, he left a will stating that "My fortune will go to the relative that avenges me." This entire branching family starts to gang up on the heroes, before The Smart Guy points out that there can be only one successor, and an Enemy Civil War breaks out. The last remaining member is too wounded to do anything, and simply limps away.