Kunya: an honorific title that begins with “Abu” (father of) or “Umm” (mother of).

Al-Hakam: one of the Names of Allah; it means the Judge whose decision, once given, can never be overturned.

“To Him belongs the judgment”: that is, He alone decides between His servants in this world and the Hereafter.

“My people … etc.”: meaning, I did not choose this kunya for myself; rather, my people bestowed it upon me.

“How excellent that is!”: meaning, reconciling people and judging between them with fairness and careful regard for justice.

“Then you are Abu Shurayh”: he granted him a kunya taken from his eldest son, for the eldest has the strongest claim to it.

Overall meaning of the hadith: The Prophet ﷺ disapproved of this Companion being called Abu l-Hakam, because al-Hakam is one of the Names of Allah and those Names must be held in reverence. The Companion explained the reason for the kunya—that he used to reconcile his people and settle their disputes to the satisfaction of both sides. The Prophet ﷺ approved of the deed itself but not of the kunya, so he changed it and named him after his eldest son.

Relevance of the hadith to the chapter: it shows that one must not demean the Names of Allah by adopting any of His exclusive Names as personal names or teknonyms.

Points derived from the hadith:
1- It proves the prohibition of debasing the Names of Allah and forbids anything that might suggest disrespect toward them, such as taking the kunya Abu l-Hakam and the like.
2- Al-Hakam is one of the Names of Allah.
3- It is permissible to reconcile disputants and to seek judgment from someone competent to adjudicate even if he is not an official judge, and his ruling is binding.