What to look for in a (Cheap) MTB headlight for night trail riding
As already suggested by @Batman, have a look at this Bike Light Database write up. 

What to look for ...
LED - with cheap lights don't believe the claimed Lumen, Look at the LED specification - a CREE XL-M T6 or U2 are probably the one you want. There's not much difference in them, you typically get about real 600 Lumen out of the cheap lights, more expensive lights with better drivers can get a bit more. IN the end, few people need more than 600 Lumen.

Battery
Lithium batteries vary enormously in capacity and price. If you buy cheap ones, expect 1/4 of claimed capacity. Buying off the likes of EBay is a crap shoot - if you want good batteries, buy locally from a reputable supply, otherwise buy cheap ones off EBay and expect to get what you pay for. Life time may be short for cheap ones, as some are known to be 'recycled' lap top batteries.  Make sure you get what is known as protected batteries. 

Construction
Heat is the enemy of LEDs, some cheap lights have plastic cases, so don't get the heat away, others have good alumina extrusions. They do not need fins when riding as the airflow is enough to keep them cool. Most cheap lights are not particularly  water proof - if you ride in the rain you probably want to go for better quality. 

Most cheap lights come with a Silicon O ring to hold them to the bars/head-mount. It works surprisingly well, but I wonder how long the O rings will last, and getting the light and battery pack on and off the bike is not a quick task.  They will not survive a big knock and a probe to breakage, and the wires are prone to breaking internally - fixable if you are handy with a soldering iron. 

Optics
Expensive lights have vastly superior optics to cheap ones. Cheap ones often have a very bright center spot with little flood - great for seeing further than you car head lights, not so great in corners, especially if helmet mounts. 

With the cheap lights on EBay (and sold in many shops), they lights may look identical, but be quite different in quality and optics. 

For a budget of $50 my suggestion is to do what I have done - Buy a couple of cheap lights off the likes of Amazon/ EBay / Dx. One for helmet, one for bars, be prepared to buy a new battery or lamp every now an then. This gives you a spare light to get you home if one fails, typically a couple of hours each on full brightness (I run both mine on medium, giving 4 hours ride time). The major down side is optics could be a lot better. Many people live with the limitations of cheap lights and replace them as they break rather than buying some good quality ones at 10 times the price.