Any type of healthy fat, be it monounsaturated or polyunsaturated, is good for kickstarting your liver, especially first thing in the morning after you wake up.
Olive oil is more well-known than avocado as a first-thing-in-the-morning pick me up for your liver, and some health fanatics are known to down a tablespoon of olive oil every morning for this reason.
Avocado can serve the same purpose because it is composed almost as much monounsaturated fat as is olive oil. In fact over 80% of an avocados nutritional composition is of monounsaturated fat.
In order to give a boost to your liver, try eating a half of an avocado a day. It is best to eat it on an empty stomach for best benefit to your liver, which is why many people choose to do so first thing in the morning.
What I do is save the other half of the avocado for the next morning using an avocado saver, which is a kitchen tool that helps to keep an avocado fresh and green by preventing the oxidation that causes the flesh to turn brown.
If you are very physically active you can get away with eating a whole avocado a day. Just keep in mind that because of their high fat content, avocados are very high in calories.