What is the best technique to photograph blue LED lights in a way that doesn't make them look over illuminated or make the background look washed out? My phone is a Motorola G Stylus and I have tried BOTH it's built-in camera app and an app called Open Camera. By example, the LED light on this Christmas tree were a nice deep blue. The picture came out looking oversaturated.

Technically, the problem is that deep blue color is so called out-of-gamut color for most cameras. Cameras (and most monitors as well) work by default in standard, but limited sRGB color space (originally designed for HDTV televisions) and typical camera/monitor pair just can not reproduce this. Also, LEDs are very bright point sources, causing dynamic range problems with cheap, and even not so cheap cameras.

How well this is handled depends on the exact camera. You can attempt to play with shooting settings - reduce color saturation, reduce contrast, adjust exposure compensation, turn on or turn off HDR. Also, you should play with scene composition, to balance how much of the light goes from LEDs themselves and how much from the external lighting. Going from wide-angle shoot to macro may help compositionally sometimes.

Looking at your picture, it seems that blue colors are rendered somewhat purplish, with some leakage into red channel. Not sure if this is a camera error, or some your postprocessing attempts. For example probing the color in the photo editor returns R=43 G=18 B=206, while R and G should be close to zero. If the camera does this, I just can suggest dumping it in the exchange for something better calibrated.
 

Deep saturated colors are just out of range for pretty much any except of some specialized camera.
It is just even theoretically impossible to render correctly all colors, especially the saturated ones. unless you have infinite amount of monochromatic channels (sensors plus display elements) to reassemble exactly the original spectrum.
The display chain is designed to render well common natural colors, like human skin, plants, wood, nature, just the common everyday things, which reflect spectrum in a smooth rather wide bands, so around the center of the color graph. They are not designed to display correctly when the color is around the edges.

So with saturated colors there will always be some color distortion. It could be better or worse, depends on how well the scene light matches the wavelengths of the camera/display chain and what are the exact color sensitivity curves of your eye. And not speaking about the exact spectrum of the illuminating light source (not valid for displaying the LED light alone, but for displaying the rest of the scene)
The camera and monitor may be standardized and there I would expect the higher quality models would correspond better to the standard, still the color may be way off with many real world LEDs (as the exact wavelength varies among LED models).
And even the same camera/monitor displaying the same LEDs in the same scene, the color "accuracy" (that means the way how the scene appear different in person vs in the photo) will vary to some extend among individual people.
So with these extremes like monochromatic light source, a camera that may seem to produce better picture for you may not be that good for someone else. Or for another LED, slightly shifted wavelength.
These are really far away from the color range the display chain is supposed to handle.