Basically, what's the difference between this:
However, it didn't mean that I didn't have the potential to become a rock star.
and this?
It didn't mean, however, that I didn't have the potential to become a rock star.
Or does it mean exactly the same thing?
The Ancient Romans knew about the power of word placement in sentences, and used it to great effect in their particular culture. In Latin, the relationship of words to each other did not depend on their position. To keep people guessing as they spoke, they often saved the most important word in the sentence - the verb - for last. The others they arranged however they liked to achieve the greatest effect.
We have lost most of that ability in English, and changing word order often means having to change the entire construction of the sentence. Your sentence has the same meaning wherever you place However. You, though, have a choice in a word you can place to your desired effect: what do you want to emphasize? "The king, they killed." has a heavy emphasis on king by disturbing the natural order of the sentence and putting king first.
If you want to emphasize something in the previous sentence, however at the beginning of the sentence does that: a sentence beginning with however will tend to draw attention to itself and its transitional function. (This is often where you will find the word. Examples will follow. Judge for yourself how placement changes emphasis.)
However, it didn't mean that I didn't have the potential to become a rock star.
(However, we haven't received your tax return.)
To suggest a contrast that is unexpected in light of the first part of your sentence, you can insert it in the middle, and it also emphasizes I:
It didn't mean that I, however, didn't have the potential to become a rock star.
(Why should we, however, have to plead ignorance...)
If you want to emphasize your potential, you can place it later:
It didn't mean that I didn't have the potential, however, to become a rock star.
(Targeting of German Chancellor Angela Merkel's phone, however, did not end until quite recently.)
As J.R. noted, you can also save it to end the sentence and emphasize rock star:
It didn't mean that I didn't have the potential to become a rock star, however.
(The term "net neutrality" didn't come into popular use until several years later, however.)
Or, in your second example, to emphasize the meaning:
It didn't mean, however, that I didn't have the potential to become a rock star.
(Another U.S. official says, however, that Obama was briefed and given detailed documents describing what's known as...)
Where you put it doesn't change the meaning. It just puts the emphasis on different aspects of your sentence.
Others have commented on the double negative, so I will not.
