Lets talk about the data first. How many binary observations do you have per year? How many years do you have? How many other variables do you have?

For simplicity lets say your binary observation is 'does it rain or not' in a given city on a given day. If you have a a continuous predictor to go along with it, lets say humidity, then you can perform a binary-logistic regression. What this would be doing is trying to fit a probability of it raining based on the humidity. Based on the model diagnostics you can determine if its a good predictor or not. Also, you can use many factors simultaneously, like humidity, temperature, barometric pressure, etc..

If you have some categorical predictors than you can include them as well into a logistic regression as well, very much like you can with linear regression. Depending on your software you might have to code the data differently to make it work. But examples would be using the month, or day of week. Again, you need to be able to interpret the results of the model to understand if the factors/predictors are significant or not.

I only proposed this one because I've used it before. I am sure some others with more experience can offer a few more ideas.