  The disagreement in the Hubble constant measured by different cosmological
probes highlights the need for a better understanding of the observations or
new physics. The standard siren method, a novel approach using
gravitational-wave observations to determine the distance to binary mergers,
has great potential to provide an independent measurement of the Hubble
constant and shed light on the tension in the next few years. To realize this
goal, we must thoroughly understand the sources of potential systematic bias of
standard sirens. Among the known sources of systematic uncertainties, selection
effects originating from electromagnetic counterpart observations of
gravitational-wave sources may dominate the measurements with percent-level
bias and no method to mitigate this effect is currently established. In this
Letter, we develop a new formalism to mitigate the counterpart selection
effect. We show that our formalism can reduce the systematic uncertainty of
standard siren Hubble constant measurement to less than the statistical
uncertainty with a simulated population of 200 observations ($\lesssim 1\%$)
for a realistic electromagnetic emission model. We conclude with how to apply
our formalism to different electromagnetic emissions and observing scenarios.
