I am not sure if "her birth waters sang like rivers" has one particular meaning, but I can help you figure out multiple possible meanings. You can go with the one that makes the most sense to you.
One possible meaning could connect to the mother's extraordinary gardening skills. The line in question resides in the second stanza. In this stanza, our narrator tells us that her mother "raises rare blooms." Perhaps her ability to grow such unique plants or flowers is what leads our narrator to liken her birth waters—which appears to be just tea—to the power of an entire river.
We might also wonder if the narrator isn't indirectly talking about her own birth. Perhaps the "blooms" are her. When she was born, that's when her mom's "birth waters sang like rivers." Though this might seem like a stretch, if we look at the ending—where Goodison says, "I...
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am becoming my mother"—we find evidence that our narrator might not be as far away from the poem as we might suspect.
Another way to think about this line is in the context of feminists who emphasize the biological power of women. We could say that linking her "birth waters" to "rivers" underscores the biological might of giving birth. As these "birth waters" sing, we should also note how this biology retains agency and a voice.
References
It's important to first consider the symbolic use of "rivers" in these lines of text in order to extract the speaker's meaning.
A river presents a bold, powerful image. It connotes a sense of change and the passing of time. The speaker links this image to that of "birth waters," or amniotic fluid. The function of this fluid is to protect the baby as she develops, cushioning movements of the growing child inside her mother's body and from any bumps the mother may endure.
When labor begins, the amniotic sac is ruptured and those fluids rush out. This is a beautiful transition, a point which marks the impending delivery of the baby whom the mother has protected and nourished for nine months.
The speaker notes that as these fluids rush out, they sing. This is a powerful verb choice, conjuring emotions of joy and elation. Just as the fluids have done their job well, protecting this infant for nine months for this very moment, the mother has also done her job well. There is therefore a "river" of emotions which rush forth in this moment, and the moment of delivery connects this mother to all mothers before her and all who will come after. Labor and delivery require a fierce strength, and her mother stands for that instant in the river of time, at once the mother of her own child and simultaneously part of the inexplicable connection to all mothers everywhere.
Lorna Goodison's "I am Becoming My Mother" is a beautiful, image-heavy, celebratory poem that describes how much the speaker has in common with her mother, how they share specific traits as well as a sense of belonging together in a powerful, long line of females. Although we can't assume automatically that authors are speaking the words of their poems directly, in this case, we can: Goodison herself has said that the poem is about her and her own mother.
Here's the second stanza:
My mother raises rare blooms
and waters them with tea
her birth waters sang like rivers
my mother is now me
The line in question is the third one: "her birth waters sang like rivers." It's a rich one, very open to interpretation!
In one sense, this line could mean that when the speaker's mother gave birth to her, the rush of the mother's fluids that had nourished the baby was like a powerful song of nature.
In another sense, the line could mean that, like a singing river, the speaker and her mother (and the rest of their female ancestors) are connected in a long, flowing, beautiful line of femininity and humanity.
In a third interpretation, you could read the line as a reference to the family's home country of Jamaica, as an indication that the land or the other people living there were joyous when the speaker's mother was born. (By calling a place your "birth waters," you might mean it's the place where you were born, the place that continues to nurture you physically and spiritually.)
Finally, you can interpret the whole stanza to mean that the speaker's mother raised wonderful children ("rare blooms") and that every time she gave birth to a child, it was a joyous, almost musical occasion. If you favor this interpretation, you'll be interested to know that Goodison had eight siblings!