Don Rubin wrote an influential paper proving that there is no single imputation method that will produce unbiased inferences (where "single imputation" means the imputing of only one value for a missing observation). However, in the same paper he pointed out that it may well be possible to create multiple imputations whose mean is an unbiased estimate of the missing value, and whose contributions to increased variance in subsequent analysis is a reasonable estimate of the added uncertainty due to data missingness.
This is his paper:
Rubin, D. B. (1976). Inference and missing data. Biometrika, 63(3):581–592.
And this an update to it: Rubin, D. B. (1996). Multiple imputation after 18+ years. Journal of the American Statistical Association, 91(434):473–489.
And this a gentle introduction to the topic of multiple imputation:
Schafer, J. L. (1999). Multiple imputation: a primer. Statistical Methods in Medical Research, 8:3–15.
There are a variety of statistical software packages that support multiple imputation (e.g. mice in R, or ice in Stata, or indeed Stata's built-in multiple imputation capabilities in recent versions).