is not a standard, recognized phrase in either Spanish or Chinese. It appears to be a combination of words from two different languages.
Here is the breakdown of the likely intended meaning:
The pen name of the famous modern Chinese poet Zha Haisheng (1964–1989.
The characters in his pen name are different from the common word for "child":
海 (Hǎi) means "Sea" or "Ocean".
子 (Zi) means "Son" or "Child" (often used as a diminutive suffix).
What this likely means when your son is telling you that he is getting a pet(animal) primero haizi he is really saying he his having his first child.
primero(first) haizi (child)