Boycott isn't the only last name that has a definition. Brown, trump, king, north, walker, wolf, grey, white, fielding, etc. If Boycott were changed then a lot of other names should be changed. And boycott isn't necessarily a negative word like wolf is, it has a very noble past, in north America at least.
No, Charles Boycott's family should not fight to get the verb 'boycott' changed. It is understandeable that they would feel the desire to get the verb thrown out, because it might feel like a stain on the family's name. However, at this point in time, the word 'boycott' is really a part of the public language. It would be a waste of time for the Boycotts to attempt to get it changed. People would use it anyways because they have for so long. Fighting to get it changed would be a waste of time and money for the family.
No, his family shouldn't fight to have Charles Boycott's name disassociated with the verb boycott. It's such an ingrained part of history, the attempts to disassociate his name would be futile. Instead, the family should work to create a different legacy, something positive or innovative that couldl be assocaited with the name Boycott for future generations.
The word 'boycott' has become an integral part of the English language, used without any reference to Charles Boycott who inspired the usage of the word. Even if his family tried to prevent people from using this word because it upset them, it is very unlikely that they would be able to stop people from doing so.
No, his family should not fight to get this changed because it is ancient history. The family should wear this as a badge of honor. Someone in their family left a legacy that has lasted generations. It may not be the legacy they wanted, but it is an interesting conversation piece when they go to parties.