If your only cavil to your family's trip to Disney World is that you don't like airline food, you're not representing the anti-Mickey side of the argument very well. A cavil is a small or petty objection.
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 about the minute, unimportant details of who owes what — like whether your portion of the bill is $10.00 or $10.17.