If your only cavil to your family's trip to Disney World is that you don't like airline food, you probably won't win the argument. A cavil is a petty objection that distracts from a true argument.
Cavil is not a word that comes up all that often. You can replace it with the more commonly used word, quibble, and still have the same meaning. If you cavil over a restaurant bill, you argue over the minute details of who owes what –– like whether your portion of the bill is $10.00 or $10.50.