In Section 7 of Alice Silverberg's [Rank "Cheat Sheet"][1], Silverberg stated

> The Bhargava Conjecture: For each $n >
> 1$ the average size of
> $S_{n}(E/\mathbb{Q})$ is
> $\displaystyle\sum\limits_{d|n} d$.

Here $S_{n}(E/\mathbb{Q})$ is the $n$-Selmer group of $E/\mathbb{Q}$. Silverberg remarks that assuming the Bhargava conjecture for infinitely many $n$, the [parity conjecture][2], and the equidistribution of root numbers of $L$-functions of elliptic curves over $\mathbb{Q}$, it follows that $50$% of elliptic curves have rank $0$ and $50$% of elliptic curves have rank $1$ (this is called the Rank Distribution Conjecture). 

> Does anyone have a conjectural
> strategy for proving the
> equidistribution of root numbers?

Silverberg mentions the Poonen-Rains conjecture together with the parity conjecture implying the Rank Distribution Conjecture (which in turn implies equidistribution of root number), so it could be that trying to prove the Poonen-Rains conjecture offers a possible approach, but it seems to me that one in fact needs the equidistribution conjecture as a hypothesis in the latter statement of Silverberg... 

  [1]: http://math.uci.edu/~asilverb/connectionstalk.pdf
  [2]: http://mathoverflow.net/questions/71609/the-parity-conjecture