Use Avocado instead of Butter to bake a cake?
There are two things you care about when you are making a cake. First, ratio. Second, sequence. 

The ratio is a weight ratio. For example, a muffin batter has 2 parts flour, 2 pars liquid, 1 part egg, and 1 part fat. You can use different types of liquid to make different kinds of muffins. You can make any amount of batter you want, provided you use the same ratio.* 

The second point is the sequence of combining the ingredients. Look at pound cake and sponge cake: they are made with the same ingredients, using exactly the same ratio. But you combine the ingredients in a different order, producing two different types of cake layer. 

If you decide to change a recipe for avocado baking, you must keep both ratio and sequence in mind. An avocado contains, on average, 15% fat and 74% water by mass. It also has 7% fiber, but as I assume that you'll puree the avocado first, it shoudln't roughen the cake too much, it will be nowhere like whole wheat cake. We can ignore the sugars (1%) and proteins (2%), as the amount is insignificant. 

As for the ratio, you can try using avocado, and then removing some fat and liquid from the cake recipe. As fresh avocado has 5 times more water than butter, it isn't possible to substitute all the butter from the original recipe - I don't know any cake recipes with such high water:butter ratios. You should rather replace the liquid, calculate how much butter is being replaced by avocado fat, and then add the remaining butter needed. It will still need fine tuning, because some of the avocado water is already trapped in the fiber, and won't hydrate the cake flour. I'd try out using maybe 140-160 g avocado puree for each 100 g liquid in the recipe. This will result in 115-120 g. avocado water. It will also deliver 20 g fat. So for aforementioned cupcakes, we'll only need 5 g additional fat (they use 25 g fat per 100 g liquid). Of course, layers for sponge cake or sachertorte are out of the question, because they don't use liquid in the first place. 

The second problem is the sequence. Some recipes expect you to cream the butter and sugar. These can't be made with avocado, as it is impossible to separate the fat and liquid in this case. Similarly, you can't use it to make a roux for a souffle (OK, that's not technically a cake). Other recipes don't require such careful mixing methods, and they should be fine with avocado puree. You can try practically anything where you are supposed to dump all liquid ingredients together, mix, and add the dry ones at once. 

Of course, you want also to consider the taste. Avocado has a somewhat earthy taste, and you'll probably notice it in a baked product. So fine cakes (like chiffon cake) are better off with butter. But most of those are made without liquid anyway. You can try to hide the taste in cakes which have a very strong taste of their own. Chocolate batters will be very good for this, but some flourless cake bases with a significant share of ground nuts should work too. Another option is to use avocado in some more hearty, heavy cakes, where the taste isn't a problem. It should function in sweet cakes, and will probably be perfect in savory cakes (think zucchini &amp; bacon cupcakes). It would be interesting to see what happens to a flaky crust where avocado puree is used instead of water (with butter accordingly reduced). If the texture remains OK, it should give a very nice note to quiches. 

If you replace a sour liquid with avocado (e.g. yogurt or buttermilk), don't use baking soda. If the recipe calls for it, use baking powder instead, because baking soda needs acidic ingredients for its leavening action. 

* But part of it waits too long for the previous patch to free the oven, the baking powder will bubble away and the muffins won't rise.