2000 Lincoln Contental armrest power outlet not working.

wrong fuse , the service manual covers all that.
aux = old cig cigar jack in cab ? <<<< names change in time
but is the same CIG port, for sure.
the manual calls it

Horn/Cigar Lighter/Power Outlet oh, so horn is dead too,, sure.
fuse 15 is front CIG
read outlet (omg 2) is fuse 6 , 50amps that feeds FUSE B
fuse B is 20 amps and feds aux rear center pin power.out.

fuse B box name is RELAY and fUSE BOX RFB.

No , can't be the relay , by looking at a wiring diagram the power outlet has different supply voltage . Fuse 1 25 amp is for power outlet an cigar lighter use's fuse # 2 25 amp. Free wiring diagrams here http://www.bbbind.com/free-tsb/ Enter vehicle info. year , make , model an engine size. Under system click on interior & driver amenity, then under subsystem click on cigar lighter . Click the search button , then the blue link. Looking at the diagram you will see they have different fuse's. Something up stream from there is blown . Need to look at electrical distribution diagrams. Go back to system an click on electrical distribution then under subsystem click on power distribution . Click the search button then the blue link. Scroll down to the # 6 diagram . You will see fuse # 4 50 amp location is under hood fuse box . Check that fuse .

AUX PWR Fuse #41
20A
Auxiliary Power Outlets - this not Cigar lighter ,the power outlet is bad. replace it or short to ground in wiring.
CIGAR Fuse #47
15A
Data Link Connector, Cigar Lighter




the power

I think you meant power outlet port under front ashtray.
Manual doesn't show specifically that port but it should be either
your left knee area when you sit on driver's seat
or
Black plastic box under the hood near battery.
You can use test light to each fuse to find which fuse is blown.
That port is not controlled by ignition (port continued to have power 24-7, even you take key out)

Is this from the cigarette lighter or the aux power outlet?
There are 2 fuse panels in the '99 Durango. The one containing the Cigar Lighter fuse is on the drivers side of the dash with the door open. 15 amp (blue) fuse, 4th from the front on the bottom row.
Aux power outlet fuse is located in the fuse panel on the inside the cab but on the back side of the panel.

some of these use a fueable link wire near the connecting block under the hood also.. be checking this too if in doubt though you can always rewire these fairly simply

It's usually the cigar lighter/power outlet fuse in the under dash fuse panel. I see a lot of people have bad cellphone/iPod chargers that blow this fuse.

check fuse 41 in the central fuse block. I bet you 12 volt outlets (cigar lighter) is not working also?

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The fuse you are looking for might be in the fuse box under the hood. I think on or about page 203 of the owners manual it will show the locations of the fuse boxes. The one under the dash is a little tricky. Just figure out what by looking at the pattern of the fuse lay out. You are looking for a 15 amp(blue) fuse I believe.

After you fix that I would recomend using the power outlet. I believe there are 2. One just in front of the shifter and the other in the center armrest storage box.

According to the Haynes repair manual: "There is no fuse and relay box inside the passenger compartment.  However, there are a number of fuses, relays and circuit breakers in various locations under the dash.  The heated mirror circuits and the circuit for the lower instrument panel power outlet (in the floor console) are fused with self-resetting fuses that are only serviceable by an authorized dealer."  However, in the fusebox under the hood (a.k.a Integrated Power Module) there is a fuse RDO/IP IGN which is the fuse for the radio.  Check this fuse.  Both the radio and the Cigar Lighter are fused through the IPM.