I came across this statement involving さえ.
私はAKB48が全然分からない。一番有名な人さえ名前がわからない。
My question is: Is there a general rule as to where you can place さえ?
What about: 一番有名な人の名前さえ分からない。What is the meaning of that?
As rapier notes, さえ roughly means "even, also" and functions in Japanese a bit like a stronger version of も. The さえ particle derives from the verb 添【そ】える "to add on to something else, to attach to something else". If you want to use this particle, it should go in the same place you'd put the も. You can even use them together, in which case the さえ comes first: さえも.
To demonstrate the meaning, let's look at your two example sentences, parsing them out and diving into the meaning.
一番有名な人[さ]{●}[え]{●}名前がわからない。
一番 有名 な 人 [さ]{●}[え]{●} 名前 が わからない。
Number-one famous[ATTRIBUTIVE]
person/people even name[SUBJECT]
understand-not
Even the most famous people, and I don't know the name.一番有名な人の名前[さ]{●}[え]{●}分からない。
一番 有名 な 人 の 名前 [さ]{●}[え]{●} 分からない。
Number-one famous[ATTRIBUTIVE]
person/people[POSSESSIVE]
name even understand-not
I don't know even the most famous people's names.
The meanings are close, but as you can see, the emphasis is different. In the second example, it comes across as if the speaker doesn't know a lot of things, and the famous name is just one of those many things.
As you may know, 「さえ」 corresponds to the word "even" in English.
The sentence 「一番有名な人の名前さえ分からない」 means "I don't know even the name of the most famous person" or "I don't know the name of even the most famous person" (I don't know whether they are correct sentences, but I think they make sense to you).
In English, people place the word "even" before the object, "the most famous person" in this case, but 「さえ」 is placed after the object in Japanese.