Help me build a silent $1,000 mini-ITX gaming PC.

Silent means no sound, and may implies passive cooling. Quiet means some sound. I can help you out a bit here, as I built myself a mini-itx gaming rig based around a Silverstone FTZ01 for my living room, but the first thing I'm going to tell you is that it is not going to be silent. No gaming rig that size will be silent. But it can be quiet enough that you won't hear it over your speakers when going full bore. Think about as loud as a modern console.

Just so I know, is this a living room PC connected to a TV, or are you going to be using this at a desk? What resolution do you plan to game at? Given your budget, I'm assuming 1080p. Also, do you have amy parts you plan on carrying over from an older PC? Like an SSD or something. And do you plan to over clock?

Since we are dealing with a $1000 budget here, I'm thinking a PC based around an i5 and a gtx 1060 or 1070, depending on how many other parts we need and if we can cut some corners. Go pick out a case; Silverstone has quite a few of them that fit your need, so I recommend starting with the RVZ01/RVZ02 and their varients. Corsair has its bulldog case as well. Google "slim tower" to find cases. Once you've picked a case and answered my questions I can help you further.

The RVZ01 can be set horizontally like a console. I have my FTZ01 (a reskinned RVZ01) laying horizontally in my TV stand.

While I have no personal experience with the Node 202, it looks like a nice case and seems to get good reviews. I've use Fractal Design cases before, and they usually make good cases. Note that the Node 202 also does not have an opening for an ODD, so keep that in mind if you want one. The node 202 is also the smaller of the two cases though, which will make it a little harder to build in and keep cool. Not that the RVZ01 will be much better in those respects though, but it's something to keep in mind.

To give you some perspective on how big these cases are, both cases are a little bigger than an Xbox One, with the Node 202 being about 1/2"-1" smaller than the RVZ01 in all dimensions.

Since you are not overclocking, it's best to go with a non-K i5. I recommend the i5-6500 as it's the best bang for the buck. I would also stick with the stock heatsink for now. You can always upgrade later if you want better cooling.

For the motherboard, you can go with a cheaper H170 board. Since you may connect this PC to a TV, you should consider getting a board that comes with a bluetooth/wi-fi receiver for connecting wireless peripherals without taking up a USB slot for an external bluetooth receiver.

So here's what we got so far:
http://pcpartpicker.com/list/tKtn7h

There's a little wiggle room with the board, as there are some cheaper boards that have wifi/bt, but that's the one I liked at that price range. If you don't want built in wifi/bt we can go even cheaper. Add a case and an SSD (I'm not sure what size you want) to that list, and we'll look into adding a video card and PSU afterwards.

So here are a couple of builds based around the Node 202 that I think will work for you.
The fist one is completely on budget: http://pcpartpicker.com/list/s7dq3F

But the GTX 1060 just came out and is hard to find, especially at a decent price. You could go with an AMD RX 480 instead (the 8GB variant, 4GB of VRAM is almost minimalist at this point), which is a little cheaper and a little more readily available, but still somewhat hard to find. For my money though, I would go with the GTX 1060 as the RX 480 isn't as powerful and runs a bit hotter.

This next build is about $150 over budget (probably around $200 over after taxes and shipping), but gets you a GTX 1070 and a more powerful PSU needed to run it: http://pcpartpicker.com/list/JNDwhq

GTX 1070s are a bit easier to find now, but the cheapest you'll find one is around the $430 range.

So here's the deal, you chose to build a PC at a time when new cards are coming out, and right now there is a feeding frenzy over these cards that is driving the price up and making availability scarce. I don't recommend buying older cards just because they are more readily available, since for the same money you can get a newer, more powerful card. Instead, if you can't wait, I recommend buying the rest of the parts now and either using an older video card if you have one or just sticking to integrated graphics (it's enough to run older games and indie games) until you can get a new card at a decent price. Don't pay more than $300 for a GTX 1060 or ~$250 for an RX 480. At most, you will probably have to wait 2 weeks to a month before GTX 1060 prices stabilize.

On a final note, you should also look into getting 2 low profile/slim (10mm thick) 120mm fans for the bottom of that case, as it will help keep the video card cool, though at the cost of some noise.

So here are a couple of builds based around the Node 202 that I think will work for you.
The fist one is completely on budget: http://pcpartpicker.com/list/s7dq3F

But the GTX 1060 just came out and is hard to find, especially at a decent price. You could go with an AMD RX 480 instead (the 8GB variant, 4GB of VRAM is almost minimalist at this point), which is a little cheaper and a little more readily available, but still somewhat hard to find. For my money though, I would go with the GTX 1060 as the RX 480 isn't as powerful and runs a bit hotter.

This next build is about $150 over budget (probably around $200 over after taxes and shipping), but gets you a GTX 1070 and a more powerful PSU needed to run it: http://pcpartpicker.com/list/JNDwhq

GTX 1070s are a bit easier to find now, but the cheapest you'll find one is around the $430 range.

So here's the deal, you chose to build a PC at a time when new cards are coming out, and right now there is a feeding frenzy over these cards that is driving the price up and making availability scarce. I don't recommend buying older cards just because they are more readily available, since for the same money you can get a newer, more powerful card. Instead, if you can't wait, I recommend buying the rest of the parts now and either using an older video card if you have one or just sticking to integrated graphics (it's enough to run older games and indie games) until you can get a new card at a decent price. Don't pay more than $300 for a GTX 1060 or ~$250 for an RX 480. At most, you will probably have to wait 2 weeks to a month before GTX 1060 prices stabilize.

On a final note, you should also look into getting 2 low profile/slim (10mm thick) 120mm fans for the bottom of that case, as it will help keep the video card cool, though at the cost of some noise.

Looking at both cases, they both seem to be built following the exact same design pattern: mobo and PSU on one side of the case, GPU on the other, with a wall in between which keeps hot air from one side out of the other. In terms of airflow, both cases are designed with vents on the sides and back for exhausting air, and vents on the top and bottom for intakes.

The RVZ01 ships with 2 slim 120mm fans, 1 in the single fan slot at the top of the case, and one in one of the 2 fan slots at the bottom of the case. As the RVZ01 is designed to use positive pressure to flush air out the sides and back of the case, both fans come installed as intakes. The RVZ01 manual recommends adding a second fan to the bottom of the case when using a video card with an open air cooler instead of a blower style cooler.

The NODE 202 has mounts for 2 fans on the bottom of the case, in the exact same spot as the ones on the RVZ01, but ships with no fans. While there is also a vent on the top of the case (again, in the same place as the RVZ01's top vent), it does not have screw holes for mounting a fan and is meant to serve as an intake vent for the fan on the CPU cooler. Being a smaller case, the NODE 202 was designed to operate without any case fans, which would make the case quieter. However, I'm not sure how feasible it is to run a powerful video card at full bore with no case fans installed in such a small case, especially if you are using a video card with an open air cooler, or have the case laying horizontally, which restricts the amount of airflow coming in from the bottom of the case. I would recommend installing 2 fans in this case (as intakes) if you go with the NODE 202.

So case design and airflow is going to be basically the same with both these cases. The major difference between the two cases is size. The NODE 202 is definitely the smaller of the two cases, though the smaller size limits your choice of coolers even more so than the RVZ01 and sets a stricter requirement on video card size. That shouldn't be too big an issue though.

Thanks to it's larger size, it is possible, but not easy, to fit a 120mm AIO water cooler in the RVZ01 (I have a Corsair H75 in my FTZ01), and it's manual even gives instructions on setting up a 240mm water cooler on the video card. There is even a variant of that case, the RVZ01-E, that fits a full size ATX PSU instead of requiring an SFX PSU. The RVZ01 also has room for a slot loading ODD, can fit more storage drives (3x2.5" and 1x3.5" bays vs 2x2.5" bays in the NODE 202), and has room for an extra case fan should you feel you need more airflow.

Which should you pick? If you can do without the ODD and the extra storage space then go with the NODE 202 since it's smaller. You will have to pay a little more attention to size when selecting a video card, but the NODE 202 should fit most cards. Just be aware that the NODE 202 may run a bit hotter though, because of it's smaller size and lack of fans. Luckily, Skylake CPUs and Pascal GPUs both run very cool, and since you won't be overclocking either, you should be ok.

All that being said, the NODE 202 is meant to compete more directly with the RVZ02, which is about the same size as it. Just putting that out there, in case you haven't taken a look at that case yet (though I think the NODE 202 looks better).