The Hebrew word mishpochah means "family". It also means "relatives" or "groups of relatives", and is even used in Genesis 8:19 for "kinds" of animals.
Meaning in Ancient Israel
In anciet Israel, the twelve tribes were considered "peoples" (amim) and within each tribe were "families" (mishpochim). This language is used often in Torah, especially to describe how each census was recorded, how the divisions of Levites were organized, and how the land of Israel was allocated by lot to "family" (not merely tribe).
The head of a household was called an "elder" (zakayn), and his household (bayeet, literally "house") was also called a "family" (mishpochah). For example, God instructs the Israelites to offer the Pesach sacrifice by "house" in Exodus 12:3-4, but when Moshe repeats that command in Exodus 12:21 he uses the word "family" instead.
Meaning in the First Century
By the first centruy, mishpochah was also used for family-like community, which was valued by both the developing Messianic and Rabbinic Jewish cultures.
The concept of a family-like community was especially developed within the disciple-master relationship. At that time it was normal for a Jewish disciple, during the years he was a disciple of a certain master, to prioritize that relationship above his family relationships. This was common in every sect of Judaism in the first century that practiced discipleship.
Historical records show that in many such disciple-master relationships, the disciples not only prioritized their relationship with the master above family relationships, but also formed communities of mutual support and caring that could replace other family relationships. These disciples would even call each other "brother", and might speak of the master or the person who introduced them to the master as their spiritual "father".
When Yeshua taught that all of his followers must perpetually be his disciples, he was also telling them to permanently prioritize their relationship with him above their family relationships (Matthew 12:50). He predicted this would cause unusual strife in those cases when a family did not follow him together (Matthew 10:20-22, 34-37) and cause unusual harmony when a family was united in following him (John 17:9-26, Luke 17:3-4, Matthew 18:15-35).
Yeshua's early followers formed such communities. They were also somewhat unusual in allowing both genders to participate: the early followers of Yeshua had "brothers and sisters in Messah" as well as calling God "Father".
Meaning for Yeshua's followers in Modern Times
The demands of Yeshua's call have not changed.
We must still prioritize our relationship with him above family relationships if necessary, and have gratitude when family who are also spiritual brethren are thus blessed.
We must form communities that have the strength, emotional intimacy, patience, and forgiveness to replace family relationships for those members who lack support from their family, whether because of their commitment to follow Yeshua or because their family members are distant or have died.
Paul's advice in First Timothy 5:1-2 is still valid:
Don't rebuke an older man, but exhort him as a father. Treat the younger men as brothers, the older women as mothers, and the younger as sisters, in all purity.