The Blackmailer can be a very powerful asset to the Mafia, because he can stop one person from talking during the day once per day. The Town relies on communication to deduce who is Mafia, so the ability to stop that communication should be feared.
Blackmailing players acting suspicious may get the Town to lynch them.
If an Investigator or Sheriff is silenced before they can reveal, they may try to get an evil role lynched by voting for them multiple times (this is unlikely to work with Secret Ballot). To head the Town off from discovering the investigating role early, it may be wise to suggest they are an Executioner so the Town doesn't give them momentum.
Advanced strategy: If a Blackmailer knows they have an investigating role silenced early, it may be wise to keep that player alive so they cannot talk to the Town through their Last Will unless a Janitor is in play.
Even after an Investigator reveals his role to the Town, it is very easy to slow him down through blackmailing him since he doesn't necessarily know who is Mafia and can't tell the Town what information he's found.
That is also somewhat true for the Sheriff to a lesser degree because he can't tell the Town who he has found who is innocent, although he can still spam the vote button for guilty.
Blackmailing a Jailor may be advantageous because he would not be able to relay information to the Town.
Advanced strategy: If you have a Framer in play, have him frame the same person that is blackmailed so they seem doubly suspicious.
Finding a Mayor early before he reveals can be a winning factor for the Mafia, since a silenced Mayor CANNOT -vote and reveal that following day. This is one of the most strongest features that a Blackmailer can bring into the game, completely shutting down one role entirely. This is only a vital plan to play out if it's late game and the Mayor has not yet revealed, preventing him from solo-voting an obvious Mafia role to the stand. Or, in rare occasions, where a Godfather is not immune to detection and and Investigator reveals a Mayor and for some odd reason he does not reveal that day, silencing him can lead to Town lynching off the Mayor.
Blackmailed person can talk during trial - (Default = Off)
When playing against the Blackmailer, try to communicate that you are silenced. With more advanced players, frequently voting to lynch someone and then cancelling that vote is a good way to signal. Experienced players will work out a communication system with yes-no questions after that.
Players targeted by the Blackmailer may speak their last words.
Even if the Blackmailer dies, his target will still be blackmailed, even if it is the Veteran.