Ah, my mistake.
In constructing an index, using geometric weights is the equivalent of using arithmetic weights of the logarithms of the values, and then exponentiating. This serves to reduce the impact of more extreme values. In practical terms, this reduces the relative weight of companies with very large market capitalization.
As Value Line says, both the geometric mean and the median have this property. In fact, in the case of returns that are log-normally distributed, the geometric mean coincides with the median.
I'm not sure why Value Line chose to use geometric weights rather than medians. Perhaps it's because geometric weights are maximum likelihood estimators, and so have all sorts of nice asymptotic properties, like laws of large numbers and central limit theorems. Or perhaps because their clients are more comfortable with geometric weights generally.
Did I answer the right question this time?