How do I configure an NLog target to only log exceptions?

I can't say that I have actually tried this, but you might be able to use a when filter and a condition to achieve what you want.

Here is an example from the condition page:

<rules>
    <logger name="*" writeTo="file">
        <filters>
            <when condition="length(message) > 100" action="Ignore" />
            <when condition="equals('${logger}','MyApps.SomeClass')" action="Ignore" />
            <when condition="(level >= LogLevel.Debug and contains(message,'PleaseDontLogThis')) or level==LogLevel.Warn" action="Ignore" />
            <when condition="not starts-with('${message}','PleaseLogThis')" action="Ignore" />
        </filters>
    </logger>
</rules>

To achieve your goal, you might make a filter like this:

<rules>
    <logger name="*" writeTo="file">
        <filters>
            <when condition="length('${exception}') > 0" action="Log" />
        </filters>
    </logger>
</rules>

The idea being that you only want to log the message if the length of the exception string is > 0. Some of the when condition examples used the NLog LayoutRenderer syntax (e.g. ${message}) and some didn't (e.g. message). I'm not sure which is correct or which syntax to use in which situation. The example I posted directly above might cause messages to be logged ONLY if there is an exception present. You should also be able to configure such that your messages to one target are logged "normally" and messages to your "ExceptionHoneypotTarget" are logged only if an exception is present.

Maybe something like this:

<rules>
    <logger name="*" writeTo="ExceptionHoneypot">
        <filters>
            <when condition="length('${exception}') > 0" action="Log" />
        </filters>
    </logger>
    <logger name="*" writeTo="file">
    </logger>
</rules>

As I mentioned early on, I haven't actually tried any of this, but it seems like you should be able to do it, hopefully similar to what showed above.

Alternatively, you could use a FilteringWrapper around your HoneypotTarget. The configuration might look something like this:

<?xml version="1.0" ?>
  <nlog xmlns="http://www.nlog-project.org/schemas/NLog.xsd"      
        xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
        <targets>
          <target name="Honeypot" xsi:type="FilteringWrapper" 
              condition="length('${exception}')>0">
            <target xsi:type="File" fileName="${basedir}/Honeypot.txt" />
          </target>    
          <target name="normal" xsi:type="File" fileName="${basedir}/Log.txt" />
          </target>
        </targets>    
        <rules>
          <logger name="*" minlevel="Debug" writeTo="Honeypot,normal" />
        </rules>
   </nlog> 

I based the FilteringWrapper example on an example from from here. The way it should work, if my configuration is correct, is that all messages will be logged to "Log.txt" and messages with a non-null exception will be logged to "Honeypot.txt".

Please note that if you want only exceptions your filter should be:

    <when condition="length('${exception}') = 0" action="Ignore" />