What is the reason behind the name "power of a test"? The use of the word "power" confuses me.
You should associate the "power" of a test with the test's ability to reject the null hypothesis. The quantity $1-\beta$ is the one minus the probability of a type II error, so $1-\beta$ is the probability that the test rejects the null hypothesis when in fact the alternative hypothesis is correct. The higher this number, the higher the test's "power" ... to reject the null.
Rejecting the null when the alternative is true is a desirable behavior for any hypothesis test. But if you haven't collected enough data, your test won't be powerful enough to reject the null, even if the state of the world is that the alternative is correct. When people mention that an experimental study is underpowered, it means that not enough data will have been collected to be able to decide between the null and alternative. Read more in this Wikipedia article.