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I once computed the entropy of the US names from SSA, and there is an increase from 8.2 bit in 1880 to 11.5 bit in 2015. The increase wasn't monotonic: At the end of the 1930ies the entropy temporarily decreased. Of course, the values aren't exact (because of the cut-off in the SSA lists at 5 babies per gender and year), but the trend is clear: Names become more evenly distributed and more informative over time. While the total number of name types seems to reach a culmination point in 2008, the entropy continues to rise after that year.