Some people (e.g. Samuel E. Martin, John R. Bentley, Alexander Vovin) argue that this の is actually (or originally) a "defective verb", distinct from genitive の. Here's Vovin's explanation, from "A reference grammar of early Japanese prose" (2003):
184.108.40.206 Defective verbs
The traditional grammar makes no mention of defective verbs, these being verbs with fewer paradigmatic forms than other verbs. There are three defective verbs in Classical Japanese, n- "to be", to "to say", and to "to be". I include below a detailed description of the existing paradigmatic forms of these defective verbs.
220.127.116.11.1 Defective verb n- "to be"
The defective verb n- "to be" has only three paradigmatic forms: infinitive n-i, gerund n-ite, and attributive n-o. Its major function is that of a copula. Samuel E. Martin was the first scholar who proposed to treat the various forms of n- as rudimentary copula forms (Martin 1988:34). In traditional Japanese grammar they are usually treated as 接続助詞 "connective particles" (Ikeda 1975:205-218). [...]
18.104.22.168.1.3 Attributive form of the defective verb n- "to be"
The function of the attributive form n-o is the same as that of its modern counterpart no in the Modern Japanese examples tomodati no gakusei "a student who is [my] friend", mei no Sumiko "Sumiko who is [my] niece". [...]
Taketori n-o okina
Taketori be-ATTR old man
old man Bamboo-Cutter (TM 29.2) [...]
Vovin argues that this is also the /no/ we see used with numbers in constructions like Shichinin no samurai ("The Seven Samurai").
I'm not convinced at this point that treating these phenomena as defective verbs solves more problems than it creates (i.e. that it isn't easier just to view them as semantic extension of the particles they resemble) -- but then, I don't have nearly as comprehensive an understanding of the details as Martin/Bentley/Vovin do. So I'm throwing this in here as something interested parties might want to read up on.
Incidentally, this analysis would be susceptible to the same counterargument as Axioplase's 犬のボビー: specifically, the objection that 犬のボビー that means "Bobby, who is a dog", but お父さんのばか means "My father is an idiot", not "The idiot, who is my father". But I think that expressions like お父さんの馬鹿 can be understood as exclamations directed at the person they describe. "You fool!" → "You fool, who art my father!" (cf "Our father, who art in Heaven"). This works even if the addressed person is not present, just as I can cry "Damn you, Mendoza!" if I discover Mendoza's evil handiwork long after Mendoza has fled town.