Is it permissible to alter the eyebrows that were left [untouched] in fulfillment of a vow for the birth of a son, and what is the expiation for this vow, and would that negatively affect the son?

Allah praised those who fulfill their vows, mentioning it as a characteristic of the righteous. However, the Prophet, peace be upon him, forbade making vows, and some scholars disliked them due to their difficulty. Your act concerning your eyebrows is either permissible or impermissible: If it is permissible due to necessity and you vowed to abandon it, then it is a permissible vow. The preponderant view is that it is binding, and you have the choice between fulfilling it or abandoning it with expiation. If it is impermissible, then abandoning it is obligatory. This is a vow to perform an obligatory act, and the preponderant view is that it is not binding. As for its effect on your child, it is not possible to conclusively assert a connection with the vow. Everything is by the decree of Allah, and fearing Allah and fulfilling His covenant are among the reasons for the righteousness and preservation of one's offspring.