I believe there may be a Bayesian-style approach to get the equations given in the paper's appendix B.

As I understand it, the experiment boils down to a statistic $z\sim\mathrm{N}_{\theta,1}$. The mean $\theta$ of the sampling distribution is unknown, but vanishes under the null hypothesis, $\theta\mid{}H_0=0$.

Call the experimentally observed statistic $\hat{z}\mid\theta\sim\mathrm{N}_{\theta,1}$. Then if we assume a "uniform" ([improper][1]) prior on $\theta\sim1$, the Bayesian posterior is $\theta\mid\hat{z}\sim\mathrm{N}_{\hat{z},1}$. If we then update the original sampling distribution by marginalizing over $\theta\mid\hat{z}$, the posterior becomes $z\mid\hat{z}\sim\mathrm{N}_{\hat{z},2}$. (The doubled variance is due to convolution of Gaussians.)

Mathematically at least, this seems to work. And it explains how the $\frac{1}{\sqrt{2}}$ factor "magically" appears going from equation B2 to equation B3.

>**Summary:** The trick appears to be a Bayesian-style approach which assumes a uniform prior for the hidden parameter ($z_\mu$ in appendix B of the paper, $\theta$ here).

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As requested in the comments, here is a plot for comparison. This is a relatively straightforward application of the formulas in the paper. However I will write these out to ensure no ambiguity.

Let $p$ denote the one-sided p value for the statistic $z$, and denote its (posterior) CDF by $F[u]\equiv\Pr\big[\,p\leq{u}\mid{\hat{z}}\,\big]$. Then equation B3 from the appendix is equivalent to
$$F[p]=1-\Phi\left[\tfrac{1}{\sqrt{2}}\left(z[p]-\hat{z}\right)\right]
\,,\, z[p]=\Phi^{-1}[1-p]$$
where $\Phi[\,\,]$ is the standard normal CDF. The corresponding density is then
$$f\big[p\big]\equiv{F^\prime}\big[p\big]=\frac{\phi\Big[(z-\hat{z})/\sqrt{2}\,\Big]}{\sqrt{2}\,\phi\big[z\big]}$$
where $\phi[\,\,]$ is the standard normal PDF, and $z=z[p]$ as in the CDF formula.
Finally, if we denote by $\hat{p}$ the observed *two-sided* p value corresponding to $\hat{z}$, then we have
$$\hat{z}=\Phi^{-1}\Big[1-\tfrac{\hat{p}}{2}\Big]$$

Using these equations gives the figure below, which *should* be comparable to the paper's figure 5 quoted in the question.
[!["Reproduction" of Cumming (2008) Fig. 5 via posted formulas.][2]][2]

(This was produced by the following Matlab code; run [here][3].)

<!-- language-all: lang-matlab -->

    phat2=[1e-3,1e-2,5e-2,0.2]'; zhat=norminv(1-phat2/2);
    np=1e3+1; p1=(1:np)/(np+1); z=norminv(1-p1);
    p1pdf=normpdf((z-zhat)/sqrt(2))./(sqrt(2)*normpdf(z));
    plot(p1,p1pdf,'LineWidth',1); axis([0,1,0,6]);
    xlabel('p'); ylabel('PDF p|p_{obs}');
    legend(arrayfun(@(p)sprintf('p_{obs} = %g',p),phat2,'uni',0));

  [1]: https://en.wikipedia.org/wiki/Prior_probability#Improper_priors
  [2]: https://i.sstatic.net/sdaFg.png
  [3]: http://octave-online.net/