I own a nice multimeter, but I bought it in the US, and those things have sharp points, so I didn't want to even try taking it though airport security. So, I left it over in the US. Here in Korea, I still have many little projects I want to work on, but I don't have a meter here and I don't know where to buy one here if it is even possible.
I do, however, have a variety of LEDs here, and basically all I ever use my multimeter for is testing for a connection anyways, so I decided to build my own.
I expected it to go really quick, but I wanted it to look good and be usable for a long time, so I spent more time on it than planned, but I like the result very much.

Here is what I did:

Step 1: Ingredients

First of all, you will need supplies. I didn't have a lot, but I made do with what I had. Here's what I used:

Tools:
Soldering Iron
Hot Glue Gun
Masking Tape
Lighter
Safety Pins
Scissors
Box Cutter

Parts:
Wire (wrapping wire)
Metal Paper Clips
3.6 v Battery and Charger Board (harvested from a broken device)
Shrink Wrap
LEDs
Resistor
Tiny Plastic Box (the safety pin box)

Lots of these things could be left out or replaced, but at the absolute bare minimum you will need wire, a resistor, and LED, and a battery. Some of these things were used simply because I had nothing better, so some of the listed parts are not 100% ideal (for instance, I recommend thicker wire).

Step 2: Prepare the Enclosure

These steps could probably be done out of order and still acheive the same results. This is just how I did it.

The first thing I did was prepare the enclosure I was using, which was the little plastic box that the safety pins came in. I used my girlfriend's hair dryer to heat the stickers so they could be removed easily (good thing I don't have a real heat gun, cause I would have used it and melted that thing real quick).
Once the stickers were gone, I used some vegetable oil and q-tips to remove the left over glue. That worked somewhat, but what worked best was heating it with the hair dryer and just rubbing it off with my thumb.

Then I decided where the charging board and battery would sit in the enclosure. The battery and charger I used was designed to be charged through USB, and had a microUSB connector on the board. I considered harvesting the USB port and replacing it with something simpler, but I have no other connectors or 5v power supplies, and the project which I want a USB port for is the reason that I am building this thing in the first place, so I didn't want to take the chance of ruining my tester; the charger is a multi-layer board, and those are like a nightmare for desoldering safely. So I decided to keep the microUSB port for charging my new circuit tester.
I used my soldering iron (improperly) to melt a hole in the enclosure at the bottom so the microUSB port would be accessible. I used the box cutter to clean up the melted plastic mess the soldering iron made.

Next, I decided where I wanted my LEDs to sit. I needed to make holes for the LED leads to go through the box, so I heated the tip of a safety pin with a lighter and used it to poke tiny holes in the plastic. It worked very well, but the safety pin cools down quickly, so it was a little tedious.

Step 3: Wiring

Once the enclosure was prepared, it was time to get the thing functional.

I wanted to have a nice thick wire for connecting to the actual probes, but all I had was crappy 30 guage wrapping wire (why did I buy that?). So, I cut three wires at the same length and braided them together. I used masking tape to hold it together while I was working on it, and then I soldered all three wires together at the ends so it wouldn't unbraid itself. I did this 2 times, once for each probe.
Then, I took the charger board and battery and located + and -. I soldered one of the braided wires directly to the ground (-) where the battery connects to the charger.
I then soldered a short piece wire to the + side (where the battery connects to the board). On the other side of that wire, I soldered on my resistor.
I then soldered the other side of the resistor onto the + side of my indicator LED and slid a piece of shrink wrap over top of the LED lead and the resistor.
Onto the - side of the LED, I soldered the other braided wire and slid some shrink wrap over that too.

I melted a small hole in the bottom corner of the enclosure and ran the two braided wires out that hole.

Step 4: Probes

I didn't want to use those flimsy wires as probes, it would be hard to be sure you have a good connection and be confusing later on. So I made some little probes using paper clips.

I selected color coded paper clips (which happened to also match shrink wrap I had) and bent them so one side stuck out. I used the box cutter to remove the plastic coating on the tips and score the back (where I would solder the braided wires). I used the lighter to heat up the plastic coating where I had scored it and the plastic pulled away revealing the metal underneath.
I then soldered one paper clip to each of the braided wires, I used red for + and green for -. I slid matching shrink wrap over the back of them to cover the ugliness of my soldering job.

Step 5: Assembly

Once the probes were attached, everything was practically finished.

I took the charging board and battery and set it in place in the enclosure. Then I hot glued it like crazy, since it will require a USB cable to be pushed into it sometimes, I figured it should be pretty strong.
I hot glued the braided wires at the point where they run out of the enclosure.
I hot glued around the LEDs on the outside.

Then I closed up the box and called it finished. I tested it and my pink LED lit up when there was a complete circuit.

Step 6: Notes

A careful observer may have noticed that my design includes 2 LEDs, while only 1 of them is actually connected to anything.
I call this one the "devil indicator LED". My testing so far has concluded that when the devil is not present, the devil indicator light does not turn on. If that one ever lights up, I know something is seriously wrong.

The color LEDs I chose were pink for the connection indicator and orange for the devil indicator. I thought they were good choices and I am unlikely to use those colors for anything else, so I didn't waste any good colors. That was my rationale at least.

Another neat effect is that the hot glue I used glows in the dark. Doesn't really help a lot, but I think it's neat.