Look a line or two above that error to see which file was last loaded, as that's where the syntax error will be. It should look something like this:
16918 [Thread-5] INFO com.fs.starfarer.loading.LoadingUtils - Loading JSON from [C:\Users\Rob\Programs\Starfarer\starsector-core\..\mods\Exigency\data\variants\exigency_destroyer_Skirmisher.variant]
I believe Starsector treats JSON files as one long line, so the "at 157" part of your error message means the mistake is at or before the 157th character in your JSON file (possibly excluding comments). Rather than count characters, you can just guess at the general area of the syntax error using that information (character 157 would probably be several lines down). Never mind that, looks like JSON errors actually do show line numbers (oops). It says it's at the beginning of line 12.
Since your error is an unterminated string, there are three things to be on the lookout for:
- A missing quotation mark at the end of a line
- Double quotation marks at the beginning of a line
- A quotation mark inside of a string, which has to be escaped with \ to avoid being considered the end of that string. Proper use of an escape character would look like this: "In the four years since \"The Incident\", the system has quickly deteriorated as [...]"
Edit: added more information on tracking down that specific error.
Edit2: log reading comprehension fail.