源 Minamoto is an 氏 uji. This is the original clan name used at the beginning of Japanese history in the Yamato court. It is proper to have a no, or "of" before the uji. In the middle ages it's said the most common four uji were the Minamoto, Taira, Fujiwara, and Tachibana.

As to be expected, as names spread out, it became confusing when the uji became too common, so people started to further identity each other by adding another part to their clan name, the 名字 myōji, usually based on a place name. For the myōji there is not a no.

Tokugawa is a myōji, so a no is not attached. On important official documents, you would use your uji, which in Ieyasu's case it is Minamoto. So you'll find 源家康. Note, by convention, you're supposed to know to attach a no to a uji even though it's not written out. Hence that is Minamoto-no-Ieyasu. Even that is not his full name.

By the way, Toyotomi Hideyoshi's 豊臣 Toyotomi is an uji given to him by the emperor, so you are supposed to have a no, that's Toyotomi-no-Hideyoshi, just that people now days don't bother (per University of Tokyo professor Hongō Kazuto, it's just convention which changes with the times and is so small in the grand theme of things that "it doesn't matter"). Hideyoshi's myōji remains 羽柴 Hashiba. Hideyoshi seem to have liked to show off the uji that he received for prestige, so uses it instead of his myōji a lot. Also Hideyoshi gave both his uji and myōji to important subjects to cement ties, adopting them into the family so to speak, so a lot of primary documents survive where persons are referred to or signs Toyotomi or especially Hashiba for their clan name.

See here for personal names.

You can also read here for some more information on Japanese naming convention, by /u/cckerberos.

For /u/ImVamcat as well.