Hello, r/AskHistorians something that I've noticed for quite some time now, but never thought to ask about it was how there were common "themes" to certain names to nobles in Japan.
For example: During the Oda clan's reign it's very obvious to see the common theme in the clan's naming process to have the prefix Nobu- for the names. So like Nobunaga, Nobuhide, Nobukatsu, Nobukatsu, Nobuyuki etc.... you get my point.
Also the clan's name will be behind it as well, such as Oda Nobuhide and Date Masamune. I'll just assume that is just their last name or whatever so no real significance here.
Anyway my main point I want answered is how come a lot of the time there are common prefixes behind a name? Do these prefixes really have a meaning to them or is it sort of like a familial tradition or both?
This practice is referred to as tsūji ("common character", 通字), the continued use of a single character in names across generations.
It is, as you suggested, just a family tradition with the specifics varying by clan (or more specifically by specific clan branch). Some clans had multiple tsūji, some didn't have any, some only used the tsūji in the name of the heir, some gave it to all sons, etc.
The practice developed in the 11th century from the earlier keiji ("lineage character", 系字) tradition. Under this tradition, fathers would give all their sons names including the same character but it differed in that: 1. that character would not necessarily have any connection to his own name; and 2. all those sons would then go on to give their sons new characters. So each particular keiji only lasted a generation.
I'm afraid I don't see what you're getting at about the role of the clan name. If it helps, Date Masamune's last name was actually pronounced "Idate" at the time.