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why not use regression on the existing entries to infer what the missing ones should be?

That would push things towards the mean... not necessarily a bad thing, but presumably later steps of the analysis will be pooling/averaging data together so not that useful.

A more interesting approach, let's call it OPTION2, would be to sample from the predictive distribution of a regression (regression mean + noise), which would result in more variability in the imputations, although random so might not what you want.

The multiple imputation approach seems to be a resampling methods of obtaining OPTION2, w/o need to assume linear regression model.

Multiple imputation simply means you impute multiple times and run the analysis on each complete (imputed) dataset so you can incorporate the uncertainty that comes from guessing at missing values into your final confidence intervals and such. How you actually do the imputation will depend on the type of variable, the amount of missingness etc. A draw from the predictive distribution of a linear model of other variables without missing data is definitely a common method, but in a state-of-the-art multiple imputation package like mi in R you can choose from dozens.

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