When is a person called a "lightning rod"?

Solution 1:

A real lightning rod attracts lightning to it. A person is said to be a lightning rod to suggest they attract something to them, most often criticism, controversy, or negative comments. Usually the thing they attract is indicated in the same sentence or nearby in the context. In the referenced NY Times article, the something is not very explicit, but they seem to be saying he naturally attracts attention to himself.

Solution 2:

A lightning rod is someone who attracts a lot of criticism, in particular in order to shield other people (like a literal lightning rod diverts lightning harmlessly into the ground). It is thus related in meaning with scapegoat, i.e. someone who is blamed for the mistakes of others.

Solution 3:

Used in this sense it means a person is a cynosure:

cynosure a person or thing that is the center of attention or admiration

The term "lightning rod" can also refer to a person or thing attracts hostile attention: "When he was president of Harvard University, Larry Summers was a lightning rod for feminist criticisms after baldly announcing that women weren't as good as men at the sciences."