The standard error of a Value at Risk (VaR) estimate is a measure of the precision of the VaR estimate. It provides an indication of the degree of uncertainty associated with the VaR estimate. The standard error of a VaR estimate depends on three key factors: the function $f(q)$, the sample size $n$, and the probability level $p$. The function $f(q)$ describes the probability distribution of the returns of the investment or portfolio. The sample size $n$ refers to the number of observations used to estimate the VaR. The probability level $p$ represents the confidence level or the probability of the loss exceeding the VaR estimate. Therefore, the standard error of a VaR estimate is directly influenced by these three factors, making choice A the correct answer.

Choice B is incorrect. The standard error of a quantile (VaR) does not depend on the standard error $s$ and variance of $q$. These are measures of dispersion and do not directly influence the precision of a risk measure estimate like VaR.

Choice C is incorrect. While sample size $n$ and $p$ are indeed factors, the square root of the error is not a factor that influences the standard error of a quantile (VaR). This choice incorrectly includes an irrelevant term.