How to choose a good mini projector?

While I agree that $50 isn't going to get you a great projector, there are 3 "different" projectors on Amazon in the $40-$50 neighborhood... I state it like that because even the most cursory examination of these items reveals that they are all the same unit re-badged for their respective sellers.

What you're going to get, assuming that you get one that works, is a projector that will probably give you a workable image up to about 65" maximum, and at a native resolution of 480p. Two of these guys claim they'll run at 1080p, but I'm curious as to how good it will look. When you skimp on cost in the projector world, you lose ground on a few key specs: (in no particular order)

Brightness Good bright lamps are expensive. Cheap lamps are not bright.
This particular unit says it has a LED lamp. This means you might get a brighter light in a smaller package, at least in theory.
Contrast Ratio This is the measure of the brightness of white compared to the darkness of black. Good contrast ratios are required for blacks to be nice and dark and therefore readable to the viewer.
This particular unit boasts a 300:1 ratio, which is not very good.
Color Rendering A variety of factors affecting the projector's ability to display the colors as presented by the source material. It isn't really a numeric spec as much as a sum of all the qualities (or lack thereof) of the projector. Cannot be described in absence of the final viewing environment, as ambient light and projection surface are critical factors. Other factors will help predict how this projector will perform, such as the color temperature of the lamp. In the case of the unit in question, the LED lamp says its temp is 16770K. That's insanely high, and will project a very "cold", blue-heavy light. We can predict from this number that the projector in question may render colors nearer the red end of the visible spectrum poorly, or that the overall rendering will be uneven. For office/meeting/multimedia presentation purposes, this may not be an issue. For watching movies, it may prove distracting. But if the viewers are not critical of these sorts of details, it may also go unnoticed.
Image ResolutionThe pixel count, in H x V. More pixels = finer image. It also means that individual pixels are less visible. As the image gets larger for a given pixel count, the image will become progressively more "grainy". Single pixels will be visible and possible distracting.
Motion Blur Depending on how fast the device generating the colored image can react, one frame may bleed into the next. Better projectors handle this problem more elegantly.

A cheap projector would, in theory, perform worse in all of these categories than a more expensive one. But, you may get lucky. At $50, it isn't an expensive experiment. If you're feeling saucy, buy one and let us know how it looks. Oh, and we love pics.

Hi, there!
You can choose a laser UST projector and Vividstorm flooring projector screen.
It is very hot and popular on Youtube.
I just bought one of 120 inch S pro floor rising screen from Vividstorm screen.
It has a very sharp and Vivid image.