One more idea I didn't see is to mark the variable as "invalid". A general "this variable doesn't have a value because the program did something bad", which carries a full stack trace with itself. Then, if you ever use that value anywhere, the result is again invalid, with the new operation attempted on top (i.e. if the invalid value ever appears in an expression, the entire expression yields invalid and no function calls are attempted; an exception would be boolean operators - true or invalid is true and false and invalid is false - there may be other exceptions to that, too). Once that value is no longer referenced anywhere, you log a nice long description of the entire chain where things were wrong and continue business as usual. Maybe email the trace to the project lead or something.
Something like the Maybe monad basically. It will work with anything else that can fail, too, and you can allow people to construct their own invalids. And the program will continue to run as long as the error isn't too deep, which is what is really wanted here, I think.