What does the following sentence, especially the highlighted phrase, mean?
In that state of union, president Obama did not so much as mention the 10 sailors that had been arrested by the pirates.
Oxford Living Dictionaries defines so much as like this:
[with negative] Even.
and gives the example sentence
He sat down without so much as a word to anyone.
The sense of the adverb even in that definition is defined nicely in Merriam-Webster:
2a —used as an intensive to emphasize the identity or character of something (forgot his car keys and even left the engine running)
So the meaning of the crucial part of your example sentence is
...President Obama didn't even mention ...
which means that the writer is stressing that President Obama did not mention the ten sailors.