The sentence is : What does my name sound like to you?
the expression "to you" what do you think it is.
If we convert into german, what would be the translation.
Will it be:
Also what does 'für dich an' means?
First of all, was is the wrong word to use. Note that English does not use what directly either, it uses like what. This is a relatively recent replacement for how. German does not have that peculiar usage, we just use wie. Next, the word that your are looking for is not hören but sich anhören. Note that this is a separable verb, which probably explains the an that you saw somewhere and did not understand. A more direct correspondence to the English sound would be klingen.
Now to get to your actual question, we use für + Akkusativ here, i.e.
Wie klingt mein Name für Dich?
or
Wie hört sich mein Name für Dich an?
