Why does the poet say that the rainbow's mother's name was tears?

There are around one hundred different species of kingfishers. Despite their differences, they have some similarities; two of those are their brightly-colored feathers and their proximity to water. They enjoy a habitat that lends itself to lots of fish, as their name suggests.
In the first line, the speaker notes that the rainbow gave birth to the kingfisher, linking the striking colors of the bird to the majesty and sense of wonder that emerges with the sight of a rainbow. The speaker then goes on to say that the rainbow's mother's name is Tears.
Rainbows only emerge after a rain or in the presence of a watery vapor. The personification here accomplishes two things. First, it helps the reader recall that things of beauty emerge after great pain, using Tears symbolically to represent rain. In literature, rain often symbolizes suffering or sadness, and rainbows often symbolize hope. While it is easy to focus on the positive aspects of a rainbow (and, therefore, hope), it is important to remember that hope is only possible because of the capacity to overcome some sadness or struggle (the Tears).
Second, the Tears remind the reader of the natural habitat of the kingfisher. I envision watching rainbows appear underneath waterfalls, just before they dump into a river below, which is only possible if I can get the sunlight to fall in exactly the right angle. This sort of mist-laden habitat is where a kingfisher would live, as well, and with a little luck and just the right angle, it might be possible to spy this rainbow-colored bird through the Tears of its surroundings.