All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
Among the linguistic meanings of the word Azaan are announcing and calling (alternative spellings: Adhaan, Athaan). Therefore, there is no harm in naming a baby Azaan because the basic principle is that all names are allowable unless they involve Sharia violations or denote an ugly or unpleasant meaning or the like, and none of that applies to the word Azaan. (Note that Aazaan, with an extended ‘a’ at the beginning means ‘ears’)
Hence, there is no religious impediment to name your baby Azaan. However, it may be unsuitable to use it as a person's name because it is not commonly used in most Muslim communities, as far as we know. There are many good names available, of which you can choose. This does not mean, though, that the name Azaan is not good. Nevertheless, it is not commonly used, as we mentioned.
Allah knows best.