How to tell an IP address with 4 LEDs?

You might also consider binary, displaying a single digit at a time. But this would require the user to know (or take a crash course on) binary.

9: 1 0 0 1
8: 1 0 0 0
7: 0 1 1 1
6: 0 1 1 0
5: 0 1 0 1
4: 0 1 0 0
3: 0 0 1 1
2: 0 0 1 0
1: 0 0 0 1
0: 0 0 0 0

To indicate the decimal point, you could show 1 1 1 1. It would be ideal if you had a button or some form of user interaction so that you could iterate through the digits.

Provide a well-mounted cord for the user to swing the device around in the air like a lasso

Then flash the LEDs like a propeller clock

_{(source: embedds.com)}

I'd do the reverse. From a control station, I would bring up a list of all IPs used by my devices. I'd then select one to start blinking in a pattern that would be easy to recognize (like 1 2 3 4 over and over) until shut off. That way I could ask everybody who's LEDs are blinking like that and know what device owned that IP.

I'd then write the IP on the bottom of the device in magic marker. There's an amazing amount of bandwidth in a sharpie.

Use all 4 displays at once for each number, showing it in binary. Blink all 4 really fast for a 0, light all 4 longer to denote a point.

[ ] [ ] [ ] [x] # 1
[x] [ ] [ ] [x] # 9
[ ] [ ] [x] [ ] # 2
[x] [x] [x] [x] # . (long)
[ ] [ ] [ ] [x] # 1
[ ] [x] [x] [ ] # 6
[x] [ ] [ ] [ ] # 8
[x] [x] [x] [x] # . (long)
[x] [x] [x] [x] # 0 (short)

Alternatively you can use an un-used number (ie: 10) to denote 0

[ ] [ ] [ ] [x] # 1
[x] [ ] [ ] [x] # 9
[ ] [ ] [x] [ ] # 2
[x] [x] [x] [x] # . 
[ ] [ ] [ ] [x] # 1
[ ] [x] [x] [ ] # 6
[x] [ ] [ ] [ ] # 8
[x] [x] [x] [x] # . 
[x] [ ] [x] [ ] # 0 

Having a lookup table ready by the device should be enough for those who don't know binary.