So he sets up a toy problem (drawing from a baby names distribution), then never explains how to solve this problem.

The intuition is, you set up a graph where the vertices are names, and the edges are based on name similarity. Two names are neighbors if e.g. their edit distance is within some limit. You start at a random name, then at the neighbors, flip a biased coin with the ratio of the P(x) of your current name and the neighbor, if heads you move to the neighbor.

I'm sure this is wrong in many and subtle ways, but when I read an article like this I expect some intuition like this to be imparted.