I would imagine the context of that statement was in the context of a cypher where a word repeated in multiple places produces the same cypher text in each location. If I see:
AER TEO ZRE SGR. AER FSD ZFD DFG. YTR ASD AER DSG.
Language analysis would reveal that "AER" is likely "The", and from there if you intercept sufficient number of encrypted messages, you can begin to determine other words from pattern of usage and create a reverse mapping. Also notice that this allows us to determine the original message of the cypher in a piece meal fashion, without ever actually "cracking" the encryption. The author's advice helps mitigate this technique by reducing repetition/patterns that language analysis might prey upon. They might figure out the reverse mapping for "the", but if you used other synonyms for "the" then it's usage in those other locations won't be as obvious.
Modern Cyphers, if used improperly, actually have this same weakness.
If I encrypt a sentence such as "The fox jumps over the bridge." using Cipher-block chaining(CBC), then the first "the"'s cypher will be different than the second "the" as previously encrypted data "waterfalls" to each next block causing identical values to almost never produce the same cypher. Thus the above attack is generally not applicable.
If however I use the weaker method of electronic codebook (ECB) then both "the"'s will(technically could, see comments) have the same cypher text, and therefore be vulnerable to similar attacks. As an example of how "patterned" ECB can be, take a look at this image encrypted with ECB:
It is possible to break a CBC down into a weaker ECB if used improperly by splitting a long message up into smaller messages. In the extreme case, consider if you split each word into it's own message, starting and ending a new CBC processing for each word. Someone on the wire would see the series of messages and notice that some would be identical because they are both encrypting the exact same message. (this assumes same key/IV used for each message/word)
Another scenario is where you use the same initialization vector for multiple small messages. The beginning of the message, and/or if the message is smaller than the IV size, then the same message (or at least the start of a message that is identical), will have the same cipher text. I've actually seen people make the mistake of chunking their message up into pieces for transfer over the wire, then encrypting each chunk separately, which produces this type of code-book vulnerability.
So Mark is mostly spot on by identifying that modern cyphers are not vulnerable to the type of attack the author's advice is trying to protect against. The reason this is the case is because modern cyphers recommend that ECB not be used (electronic code book).
However, should you be using modern encryption in a way that causes it to break down to being essentially a code-book, then you could the mitigate risk by heeding the author's advice of varying synonyms. So the advice is somewhat applicable, but that's more of a weak bandaid compared to fixing your encryption method such that it is not a code-book method.
To address some of your revision's/comments: What the author is referring to(a mitigating protection against code-book vulnerabilities), and the idea of NSA flagging certain words , are really two completely orthogonal concepts. The former is dealing with code breaking and protections against code breaking. The latter is speaking under the assumption that the NSA has already accessed the plaintext of the message, and isn't trying to break any type of encryption, but instead simply identify "messages of interest".
With ECB, "dog" and "puppy" will have extremely different cypher text, so any similarity between the words disapears post-encryption. If "dog" was used in 20 places in a ECB encrypted message, it's cypher text would appear in 20 places. If you instead used a slang list of 20 different words for dog, then you'd have 20 different cypher texts and it would not be obvious that they were related. This is the concept I believe the author had in mind.
On the other hand, the technique suggested by the author "might" work for avoiding flagging by a "message of interest" scanner, but for very different reasons than the original author was suggesting this technique for. In this scenario, the technique will have almost no effectiveness because if the NSA is using anything on the level of sophistication that Google uses for its search engine, it will easily identify similar words and synonyms. There are already lots of algorithms for analyzing the "distance" between two words to identify misspellings, and there certainly are readily available synonym lists.