What makes a person deserve to have a son be picked by HaShem as a prophet? Why was Jonah ben Amitai chosen?

The Rambam writes that prophecy is earned by the navi himself- and not through his father or mother and not chosen before one's birth (with, if I remember correctly, one exception: Yirmiyahu). It's earned by a high level of midot tovot (character traits) and Torah wisdom. On the other hand, some commentaries do relate to Amitai, who's name is from the word "truth", and that inevitably like all people, a navi is also influenced by his parents and upbringing, which helped enable him to reach those high levels of midot & Torah! Also, the general rule is that if a prophet's father's name is mentioned, that infers that he was also a prophet (Megilla 15a).