I want to create a trigger function that can set a value for a column if the column was not specified in the update statement. [...]
However if the row to be updated already has a value for last_updated_by even if the last_updated_by column is not specified in the update statement, then the "IF NEW.last_updated_by IS NULL THEN" is never fired..
Thoughts?
Not specifying a column in an update is shorthand for "leave the column unchanged". There isn't a way for you to make it mean something else - you only get to see that OLD and NEW are equal.
Thus, your requirement is not possible as described. And if you have to allow for arbitrary update commands your requirement is going to be impossible to enforce. Otherwise, you could make all updates go through procedures/functions and enforce the requirement there.