Get a GV-USB2 capture card! It's basically the go-to for standard definition console gaming.
He'll also need a video/audio splitter. I use this one and it works wonderfully. You'll need an extra set of composite cables as well. (That's just an example. Get whatever will ship on time for you)
If he wants to stream on twitch, consider getting him a webcam and mic. Let me know what your budget is and I can recommend some options for you.
You're gonna need a splitter- Signal comes from the console into the splitter, then a line from the splitter goes into the CRT TV and a line goes into your capture card.
https://www.amazon.com/gp/product/B010EIK6K6/ref=oh_aui_detailpage_o06_s00?ie=UTF8&psc=1
This is one of the more commonly used splitters. If you look in the "Frequently Bought Together" section, you'll see some extra cables you'll need. There's also a I-o capture card in that section. It's a small, cheap japanese capture card that's super easy to set up and use, and comes highly recommended for streaming RCA consoles. If your setup with the RCA to HDMI converter doesn't work or seems laggy for any reason, you might want to give that one a try.
EDIT: In addition, DO NOT BUY a CRT TV online. Online sellers have caught on to the fact that CRTs are fashionable as "retro gaming TVs", and have been jacking up the price. If you don't have one, ask around. Check craigslist. Go to Goodwill or a similar "junk I no longer need" store. You should be able to get one for next to free. Another side note is that some TVs have a "Game" or "Input" mode that there is no button for on the TV. You'll need a universal remote from Walmart/Best Buy/etc to access it. In general with old TVs, the more recognizable the brand name, the better.
EDIT EDIT: With the proper settings, MANY games will not give you problems playing on a flatscreen modern monitor. That said, if it's a game that requires a lot of precise execution/has anything pixel or frame perfect, you're still better off with a CRT.
It really depends on how deep you want to get into it. The basic composite/S-Video capture devices on Amazon/eBay/AliExpress do work if you want something really cheap. Combine that with a distribution amp so you can play on a TV while you capture. Or if you want a higher-end setup, a RetroTINK-5X plus a component video cable and an HDMI capture device will get you the best possible quality from original hardware.
You need a video capture device. There are dozens of options to choose from, including standalone ones that can record to an SD card.
If you have the budget, I highly recommend using a RetroTINK-2X since it makes getting good results much simpler. You'd record the output from that using an HDMI capture device. Some have HDMI passthrough, but for others you'll need an HDMI splitter so you can also connect it to your TV. Or if you need to be able to play on a TV with no HDMI input, you can use a distribution amplifier to split the composite video before the RetroTINK-2X.
Slightly cheaper on eBay. Don't know of any cheaper ones particularly worth recommending, though the basic EZCap clones on Amazon will work in a pinch. You'll also need a distribution amp if you want to play on a TV at the same time.
Do you want to run it to multiple tvs? You could have a powered distribution amplifier. You plug in what you want to play in the front (in my case, the s-video output of my switch box), and have it wired to up to 4 TV's with individual s video and audio cables, and it sends the signal to all of them. I have this one and it's worked for s video for me. To my eye it seems to truly send the s video through unhindered, I don't think it's converting it to composite and sending that on the s video line but maybe I just can't tell the difference. I kept the cable lengths pretty short, and since it's powered I don't think there's enough voltage drop to degrade the signal either.
https://www.amazon.com/RadioShack-4-Output-Component-Distribution-Amplifier/dp/B010EIK6K6/
You need a capture card. This is a popular one: https://www.amazon.com/dp/B00428BF1Y/
To play on your TV and capture at the same time, you need a distribution amplifier: https://www.amazon.com/dp/B010EIK6K6/
For a significant improvement in video quality, consider using S-Video instead of composite video: https://www.ebay.com/itm/132861494962
You'll also need two sets of male-to-male cables, one for the capture card and one for the TV: S-Video or composite
Also, for what it's worth, I've never heard of someone getting banned from Twitch for streaming emulators.