Avocado oil is very rich and emollient. Try to get expeller pressed oil if possible (rather than heat or chemical extracted oil) because expeller pressed is more natural and therefore healthier.
I like avocado oil, but I also adore:
-Coconut Oil (which is actually a solid until it touches you skin or gets warm)
-Rose Hip Seed Oil
You can even use Olive-Oil (extra virgin, expeller pressed is best), which is very mositurizing. It will give you an idea of whether or not oil will work well for you as a moisturizer.
However, one problem with using just oils is that they come off very easily if you eat, drink, or lick your lips. Even if you do none of these things, just plain oil will be gone in a fairly short amount of time.
A lip balm made with your favorite oil and some beeswax (totally natural) will stay on your lips and mositurize them longer because the beeswax helps the product stick to your lips. But read the label carefully and make sure there are no nasty chemicals in the product.
Actually, any type of mositurizing anywhere on the body or face is more about 'keeping moisture in' than it is about 'putting moisture on'. In light of this, using something that does not wear off easily is best, especially for lips.
After buying, trying and making hundreds of lip products, I have finally come to the conclusion that my favorite, that works best for me, is plain old petrolatum (petroleum jelly - - aka vaseline). Or a mix of petrolatum and a tiny bit of beeswax.
If the idea of petrolatum bothers you, in the US and Canada there is a product called "Un-Petroleum Jelly" made by Alba, which works just like petroleum jelly, but is made from natural oils and a touch of beeswax. You can get it at Walgreens, at the health food store, or online at Amazon and hundreds of other sites.