This shows that the 9 Volt D.C. battery works.
The LiIon battery at 3.7 V D.C. does not.
I plan on making a "fake burglar alarm" for my car.
It will be mounted in a black project box.
I already have a built factory alarm.
I would like to use a USB cable from an outlet in my cigarette lighter in my car.
A USB cable is about 5.0 volts D.C.
Any other ideas ?
Flashing LED's typically run on 3V-5V DC.
It seems that your model needs just a little bit more than 4V DC or your Li-ion battery is flat. 
Could you check with a voltmeter if the LED is actually getting something between 3 and 5V DC from the Li-ion battery ?
By the way, the reason it doesn't blow up on your 9V DC test is because of the internal resistance of the battery.
If you have the make and model of the LED on hand, I could give you a definite answer.
Without it, it is just an educated guess.
Based on that, I would say that it might actually run without problem on 5V DC.
Feeding from your cigarette lighter 5V adaptor would indeed be the easiest way.
Below you'll find the pinout for USB-A and mini-USB. You'll probably need it.
His largest problem will be that with most all late model cars and trucks the USB and CIg lighter power ports are de-energized when the ignition is off. Find a place to tap off 12 volts when the ignition is off and put a 3.3K to 10K resistor in series with the LED. Lower resistance means more brightness. Insure that the connection is fused, either before where you tap it off, or add a fuse in series with your wire to the LED.
