I'm fairly well read in Japanese history, esp. concerning the Sengoku era, but have yet to find good sources explaining the meanings of the majority of Japanese names. I am aware of the commonly fluid nature of names in feudal Japan (e.g. Tokugawa Ieyasu changed his clan name from Matsudaira), and am also aware of the meanings of common given names like Ichiro (一郎) for first son and Jiro (次郎) for second/latter son.

But what about the meaning of names like Takeda (武田) or Hojo (北條)? I speak Mandarin, and reading those kanji characters would give the meanings "martial field" and "northern stripe" respectively. Did the Japanese take into account the individual kanji meanings when adopting their clan names? If/if not, what's the process an individual would go about selecting characters for his/his clan's new name?

For example, how specifically did the former Matsudaira Takechiyo decide on the characters 德川家康 ("virtue stream family health") to refer to himself? Was there a formal procedure that led him to eventually choose the syllables "toku" + "gawa"?

edit: formatting