SeaDweller,
If the brown wire with the black(?) trace is source voltage, then the brown wire (your arrow) is ground (common ground). If you have a voltmeter, ground the black meter lead and connect the red lead to the cig. lighter terminal (brown/black wire). This wire should always be "hot" so the ign. key can be ON or OFF for this check.
If available voltage is okay (12 v), disconnect the fuse for this circuit, then use your ohmmeter (same terminal) to check for resistance/continuity to ground with the lighter pushed in. If no continuity or infinite resistance, verify you have a good ground connection, then replace the cig. lighter assembly. Basically, you want to verify there is a current path through the lighter to ground.
BTW, my lighter assembly provides a lousy connection to ground which results in an iffy connection to my cell phone charger.
Hope this helps,
Sherwood Lee
'69 911 "rs"
http://members.rennlist.org/911pcars