What determines why any particular soul is placed within a specific person or family? Does the soul have any choice in the matter? Is there any way that parents can somehow choose what soul is placed in their child? Might the placement of souls have anything to do with reincarnation and the set of challenges a person is confronted with regarding family or in life in order to achieve its “tikkun” (rectification)?
This is a very interesting and stimulating question!
The simple, general answer seems to be that in most cases, practically speaking, it is G-d alone who is directly and solely responsible for deciding which souls go to which bodies, thereby determining to a large extent everything that will happen to the soul, based on the context in which G-d, for reasons known only to Him, chooses to place the soul in this world.
That being said, there are some fascinating exceptions that I can think of which relate to your questions.
The soul generally does not choose a body. In fact, in most cases, if the soul could choose, it would choose to remain in the spiritual plane with all its delights and splendor in the presence of G-d, and not be confined to a course and limited body imprisoned in the physical plane. G-d therefore “forces” the soul into a body in this world for its ultimate good. However, G-d does not force “righteous” souls this way, since they do not need this rectification. Nevertheless, such souls “agree”, i.e. choose, to be placed in bodies in order to benefit the world in general and help others fulfill their potential. In such cases, the soul chooses, at least in general terms, to be in a body.
Parents generally cannot choose the soul of their children. However, parents can have a major influence on what soul G-d decides to place in their charge. Torah sources are replete with teachings indicating that the degree of holiness of the parents, or lack thereof, has a major impact on which souls are placed in their children. This is particularly so regarding a couple’s behavior and intentions during their union. In fact, there are stories of particular tzadikim who actually engaged in mystical elevations and personally chose the soul to be brought down.
The placement of souls is not dependent on reincarnation, since G-d places “first-time” souls in bodies, where reincarnation is not relevant. Nevertheless, because kabbalistic sources teach that nowadays there are no “new” souls, but rather most people are reincarnations, that would mean that, practically, reincarnation will be a major factor in determining in what bodies, and into what families, souls will be born. This will obviously have a lot to do with one’s familial relationships, their challenges and tikkun-rectification.
One very interesting expression of this involves the Torah mitzvah of yibum. If a man dies leaving a widow with no children, the deceased brother is required to marry the widow in order that their son perpetuates the deceased brother’s lineage. Ramban writes that this happens because the soul of the deceased is reincarnated as the son of the yibum couple. Based on this, the author of Pele Yoetz actually remarks that people who knew the deceased brother while he was alive are required to inform the son what he needs to rectify based on what they know of him from his previous reincarnation!