How many ways can one paint the edges of a Petersen graph black or white?

Addendum Mar 24 2016. A much improved solution to this problem is at the following MSE link which renders this thread obsolete.

Using Burnside's lemma is the right idea.

Each element of $S_5$ determines a permutation of the 15 edges of the Petersen graph. If this permutation has exactly $r$ cycles on edges, then it fixes exactly $2^r$ 2-colorings of the edge set. If $a$ and $b$ are conjugate elements of $S_5$, then the permutations of the edges they determine have the same cycle structure. (To proof this, observe that the induced permutations are conjugate in $S_{15}$, and hence have the same cycle structure.)

So you just have to compute $r$ for one element in each conjugacy class, which is mildly tedious at worst, and then apply Burnside.