"Мама взяла моего племянника..." (or rather "Отвела моего племянника...")
"my nephew" here is not Genitive, but Accusative, it's just that in this case the two forms coincide (i.e. the same). "Взять кого-то" or "Отвести кого-то" is the same as "Видеть кого-то" (Accusative).
As regards "story hour", phrases of that type (i.e. when nouns are used as adjectives) are often translated into Russian using Genitive, because in Russian we can't turn nouns into adjectives so freely as you do it in English. You might translate it as "исторический час" using a real adjective, but that would rather mean "history hour", not "story hour". But sometimes it can be done all right, e.g. "stage actor" = "театральный актер"(literally "сценический актер" though we don't usually say it like that or "актер сцены", Genitive again, but this is a less common expression). So the thing here is that we always have to use real adjectives, which might not be available, or if they are technically available, they might confuse the meaning not being quite the equivalent of similar expressions with Genitive.

"This is your idea"="Это твоя идея" is Nominative, not Accusative.
"This is your nephew"= "Это твой племянник" - the same.
"I took your idea" - "Я взял твою идею" is Accusative, as well as "I took your nephew".

Once again, the pronouns my, your, etc. don't produce Genitive themselves, they just have to be used in Genitive when the nouns they describe are in Genitive, be it because of the possession meaning or for any other reason, i.e. depending on the circumstances such pronouns can be used in any case from Nominative to Prepositional, since their form depends on the form of their nouns, the only exception being "его" (his) and "eё" (her) which never change their from and are the same whatever the case of the noun.
"This is his idea"="Это его идея" (Nominative)
"I took his idea"= "Я взял его идею" (Accusative)
"I'm thinking of his idea"="Я думаю о его идее" (Prepositional), etc.

Another important thing is that showing possession is not the only function of Genitive.