I am in the planning stages of moving a cold air return from an inconvenient spot in the floor to a nearby wall. The basic plan is to open up the wall on both sides (cutting one stud and framing in a filter grille box) and cut a smaller hole in the floor on the opposite side of the wall from the filter – the "back" of the wall.
To make this work, the incoming air will need to make a 180º turn from where it passes through the filter to where it enters the main return flex duct in the basement (16", if that matters). We don't have a lot of room on the back side of the wall, so need a compact solution – <14" outside the back side of the wall – that won't create excessive noise. The system specifies 850 cfm.
What parts/products should I use to route the air like that? I know there are companies that create custom returns, which I'm open to if the price is reasonable, but is there an off-the-shelf solution? Is this just a bad idea to start with?
— Edit —
Here's a sketch: 
Best Answer
If you want to go completely amateur, you could just stuff it all in there. However, that's going to result in a significant decrease in airflow and will result in a lot of strain on your air handler's blower motor.
The cross-sectional area of at 16" ID duct is about 200 square inches. That's the number you need to match or exceed to maintain the same possibility of airflow.
Since you are using a rectangular vent and you only have 14 inches of depth, let's figure out how big of a rectangular duct you need. 200 / 14 = 14.3 inches. 14x14? That's pretty close to some of the standard sizes, like 10x20. But since you have to make 2 90 degree bends, up-sizing is a good idea. Also, a larger grate can help reduce airflow noise. So I would go for 12x24, which is almost 50% larger.
So at your local home improvement center, you can try and get a 12x24 filter grate, two 12x24 90 degree rigid metal ducts, and a 12x24 to 16 ID adapter.
However, it's likely you will not find those exact pieces. So what you can do instead is get a few sheets of duct board and make a duct. Duct board is like a rigid piece of insulation rated for HVAC airflow. You will also need a 16 inch collar to attach your flex duct to your new box.
In your case, I would make your duct box extend down through the floor (where you have your two question marks) and attach your flex tube to the face of a duct board. This way you minimize the number of bends you put in the flex tube.