Why do small sample sizes give more significant p-values for rare events? Maybe I'm getting a bit confused, but I have a single binomial variable with very low frequency (e.g. 1 event in 100,000 trials), and I'm trying to figure out what sample size is required to show a significant (p<0.05) deviation from the expected frequency.  If I run a billion trials I am more likely to see several events by the null hypothesis than if I run fewer trials.  And if I see an event with a single trial, it is more significant than if I see an event with several trials.
Does this mean small sample sizes are better when frequencies are low?
How does one calculate what sample size is required to show a significant deviation from a low frequency in a single binomial variable?
 A: No, larger sample sizes have more power. 
If you have something that occurs 1 in 100,000 times then, unless you have a very large number of trials, you are unlikely to see any events. And, if you see one event, you will have a very poor idea of how likely events are.
The problem is that you don't know (prior to the test) how likely events are, exactly.  You might know they are very rare, but you  won't know how rare in your  population.  If you did know that, exactly, then you wouldn't need to run the test.
Estimating proportions of rare events is not simple. A seminal article on this is Agresti and Coull. After reading that, you can also look over the Wikipedia entry which is much more up-to-date and has lots of links.
A: First, sample size calculation is not done to force a $p$-value $<.05$, but in fact to increase the power of your testing. Intuitively, the more (good) samples you get, the more representative your sample is of the population, so the more accurate your results.
In every statistical way, larger samples are always better.

If I run a billion trials I am more likely to see several events by
  the null hypothesis than if I run fewer trials. And if I see an event
  with a single trial, it is more significant than if I see an event
  with several trials.

Yes, if you run a large number of trials, your expected number of incidents under the null hypothesis of $H_0:$ probability of incident = 0.00001 is going to be larger.
And yes, if you see an event with a single trial, then it is likely that your null hypothesis is incorrect. But remember, you are not forcing the test to show that your null is incorrect. You want the test to indicate the truth, which means you need to understand more about the truth. Which means you need more samples. If in fact the incidence probability is larger than .00001, then when you get more samples, you should see larger incidents. 
Thus, a single trial might freakishly help you reject the null hypothesis, but if in the trial you do not see an inference, you will probability not reject the null hypothesis. However, if you get more samples, you will be able to (with some power) reject or fail to reject the null hypothesis.

How does one calculate what sample size is required to show a
  significant deviation from a low frequency in a single binomial
  variable?

Let the $H_0: p = p_0 = .00001$. Lets say the 95% confidence interval you make for $p$ you want at most of half-width $a$. Your test statistic will be
$$\hat{p} = \frac{\text{# incidents}}{\text{sample size } n}. $$
The variance in the statistic under the null is
$$ Var(\hat{p}) = \sqrt{\frac{p_0(1-p_0)}{n}}$$
So you want $1.96* \sqrt{\frac{p_0(1-p_0)}{n}} \leq a$
This is the general method used for sample size calculation. There is a separate field for rare-event analysis, and it is possible that calculation of the sample size is entirely different. But this is the standard process.
