I just want to throw out there, as someone named John; he is giving us a bad rap here.
I also want to say I don’t like how close “Cavil” is to “Cavatar” in spelling.
If I were a Cylon I know I wouldn’t out myself, JOHN CAVIL. We’re onto you!
I love it. Cavil hates his name presumably because it is so ordinary. Again, his last name means ‘petty’. He is afraid of being ordinary and is therefore petty about it. Ellen described his emotions as jealousy and rage, as being the most base.
I like the name John. Why was Cavil so mad anyway? Oh well he is bitter old dude, enough said.
If you have a “pet” Boomer you have no right to complain!
He’s probably rejecting the very fact that he was named by her, regardless of the name… in mythology, when you know someone’s name, you have power over them… John pretty much said in his monologue that he wants to be a god, so he’s rejecting all mental holds that Ellen has over him, including his name.
JC also spells John Connor 
cavil
1.to raise irritating and trivial objections; find fault with unnecessarily (usually fol. by at or about): He finds something to cavil at in everything I say. –verb (used with object)
2.to oppose by inconsequential, frivolous, or sham objections: to cavil each item of a proposed agenda. –noun
3.a trivial and annoying objection.
4.the raising of such objections
the root word in latin means to deceive. That sounds like our boy, doesn’t it? 
Sure enough does at that.