Median may be [more robust][1] than mean, but this does not make it robust for such extreme cases like sample consisting of single datapoint.

Let's get more general for a moment, $p$ [quantile][2] (median is $p=0.5$ quantile) of distribution is the value $x$ such that

$$ \Pr(X \leq x) = p $$

Sample quantile for ordered sample

$$ x_1 \le x_2 \le \dots \le x_n $$

then median is middle value of such sample, where we have multiple definitions on how to define it precisely (Hyndman and Fan, 1996). It is convenient to define sample median for sample of size one as the data point that you have, but notice that single data point at the same time is also the minimal $x_1$, or maximal value $x_n$ of this sample. 

With larger sample size you could estimate [confidence intervals][3] for [median][4] to have better understanding of uncertainty of the estimate.
However since the intervals are defined in terms of order statistics this won't be helpful. What you can do with single value is to compute [confidence intervals for mean][9], e.g. $95\%$ interval would be

$$ x \pm 9.68 |x| $$

As you can see, the intervals are pretty wide, but also the situation is extreme, so our uncertainty about mean is also greater. If you can make any distributional assumptions you could get more precise. If you have *a priori* knowledge about your problem you could try Bayesian approach with informative priors.

Another thing that you can do is to find sample size that enables you to estimate median with enough precision. If you think of this problem the other way around, we want $np$ out of $n$ values of $X$ be smaller or equal than $x$ where $p=0.5$. This makes $X$ follow binomial distribution with parameters $n$ and $p$. [Wald confidence interval][5] for $p$ can be easily calculated as $p \pm z_{\alpha/2} \sqrt{p(1-p)/n}$ (but see [here][6], [here][7], and Brown and DasGupta, 2001). The interval [can be used][8] to find such value of $n$ that makes the interval acceptably narrow, i.e. to assess your confidence that $p$ is close enough to $0.5$ (that it is really a median). (Notice that for $n=1$ the confidence interval yields improper values that fall beyond the $[0,1]$ interval for $p$ and Wald's method should be used for samples of at least five.)

---

Hyndman, R. J., & Fan, Y. (1996). [Sample quantiles in statistical packages.][10] *The American Statistician, 50*(4), 361-365.

Brown, L. D., Cai, T. T., & DasGupta, A. (2001). [Interval estimation for a binomial proportion.][11] *Statistical science*, 101-117.


  [1]: https://stats.stackexchange.com/questions/2547/why-is-median-age-a-better-statistic-than-mean-age
  [2]: https://en.wikipedia.org/wiki/Quantile
  [3]: https://stats.stackexchange.com/questions/122001/confidence-intervals-for-median
  [4]: https://stats.stackexchange.com/questions/21103/confidence-interval-for-median?rq=1
  [5]: https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval#Normal_approximation_interval
  [6]: https://stats.stackexchange.com/questions/105972/averaging-binomial-confidence-intervals
  [7]: https://stats.stackexchange.com/questions/82720/confidence-interval-around-binomial-estimate-of-0-or-1/82724#82724
  [8]: https://stats.stackexchange.com/questions/165035/number-of-samples-needed-in-monte-carlo-simulation-how-good-is-this-approximati
  [9]: https://stats.stackexchange.com/questions/157582/what-can-we-say-about-population-mean-from-a-sample-size-of-1
  [10]: https://www.amherst.edu/media/view/129116/original/Sample+Quantiles.pdf
  [11]: http://projecteuclid.org/download/pdf_1/euclid.ss/1009213286