What does the "も” mean in "ーてもいいですか”

As you have indirectly mentioned, this can be considered as a construction of て + も.

［接助］《接続助詞「て」＋係助詞「も」から》動詞・形容詞と一部の助動詞の連用形に付く。ガ・ナ・バ・マ行の五段活用動詞に付く場合は「でも」となる。

There are several definitions listed under this entry, but I think the first definition is the most appropriate:

未成立の事柄を仮定条件として述べ、その条件から考えられる順当な結果と対立する内容の文へ結びつける意を表す。たとえ…したとしても。「失敗してもあきらめはしない」「煮ても焼いても食えない」

Translated to English, ても "expresses the intent to state a hypothetical matter that is not yet established and connects it to a sentence that is in conflict with the possible sequential consequences of that condition."

Now, for the function of も: I think も can be dropped especially in casual conversations without affecting meaning. 大辞林 第三版 says this:

「…てもいい」「…てもかまわない」など、許容を表す言い方に多く用いられて）「て」を強める意を表す。

In other words, も simply serves to emphasize て.