In terms of deaths caused, mosquitoes are undoubtedly far deadlier than any species humans have eradicated to date. In North America they are more of a nuisance than a deadly threat, but the same does not hold true for large portions of the world where they are a major disease vector. They are unquestionably worth eradicating if it ever becomes feasible to do so.
The question should not be whether or not we should eliminate the species of mosquitoes that feed on humans. Instead, we should ask how can we minimize the impact of eliminating these species. Can other species of mosquitoes that are not disease vectors, or other insects entirely, fill in the vacated niche? Is it possible to eradicate mosquitoes in test areas to see what the long-term impact is? If we do eradicate them globally, how long would we have to "hit reset" by hatching and releasing stored eggs if unforeseen consequences make it necessary?