Why does somebody argue that the number of bootstrap replications should not be a multiple of 10? At a recent conference somebody claimed that the size of the bootstrap replications should always be 999 rather than 1000.
Which argument supports this claim?

I believe that the comment is in reference to the number of bootstrap samples rather than to the size of each sample. It might simplify estimates of some common percentiles from bootstrapped samples.
Davison and Hinkley argue on pages 18-19 that

if $X_1,...X_N$ are independently distributed with CDF $K$ and if $X_{(j)}$ denotes the $j$th ordered value, then
$$\text{E}(X_{(j)}) = K^{-1}\left(\frac{j}{N+1}\right) .$$
This implies that a sensible estimate of $K^{-1}(p)$ is $X_{((N+1)p)}$, assuming that $(N+1)p$ is an integer.

So if you want to estimate the 2.5th and 97.5th quantiles to get a 95% confidence interval for some value, take 999 bootstrap samples, put the values in order, and select the 25th and the 975th.
I suppose that avoids choosing among the multiple ways of estimating quantiles.

Davison, A. C. and Hinkley, D. V. Bootstrap Methods and their Application, Cambridge University Press, 1997.
