Actually it's fairly simple... FX6000 < FX8000 < i5 < i7... So if you can't afford an i7 on 2011 socket, go for the cheapest i7 on 1151. If you can't afford that - step down to an i5... And so on and so on... If your choice is the dual core i3 or the 8 core 8300 i would recommend the 8 core...
I would say get the cheapest i5 you can get your hands on... Actually if you ask me - i would say go 5820K and forget you ever needed a CPU for whatever. No botleneck, no nothing. But if you want to go the bare minimum - get the cheapest i5 you can find. AM3+ is really old now and lacks a lot of features... Just get an i5 and you will be set.
That's why i said the cheapest i5 you can find... The thing is, theoretically the 8320 for example should be better. But the newer i5s have first of way faster cores and second - way way improved memory controllers and other communication modules improvements, that are lagging behind in the FX. Just don't step down to i3. Please, for the love of God, don't buy dual core in 2016 for streaming...
Couple of things: 1) If your brother is new to Twitch or not a very big channel he's not going to be able to upload anything over 720P @ 30 FPS. Not sure about YouTube. 2) Streaming with a i5 isn't a great idea. If he is streaming demanding games like GTA V or even Skyrim Remastered, you will notice some performance degradation (the games will "chug" a bit). It's also pretty noticeable when you are playing on-line games. Those that need a fair bit of CPU power.
I'd say get whatever works for you on a budget. Since you haven't laid that out (not sure why you are reluctant to spell out a specific dollar amount, but hey that's your call) For a entry level streaming box, I'd get a FX 8350, load it up with as much RAM as you can get (8GB minimum) and have a decent GPU (RX 470 seems like a good GPU for the money.) You can offload your encoding to the GPU in OBS, and if you go with Intel and use a CPU with a integrated graphics chip (iGPU) you can set it up to have THAT do your H264 encoding while your CPU focuses on rendering the game (your can only do this if you use the iGPU exclusively for stream rendering, and you need a dedicated video card). I've only gotten it to work WELL in Windows 10. The trade off is if you use the iGPU the quality isn't as good, and you'll need more upload speed.
you could get a x79 mobo and a 26XX series xeon used for around 300-400 aud maybe a bit more after shipping. it'll be older, but 6-12 cores with hyper threading. somthing like
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wendell did something similar with a dual xeon build
Yeah well with the upcomming black friday and cyber monday, it might be interesting to look at those 2 cpu´s. You maybe might find some good deal on those. 6 cores 12 threads overclockable. i think a realy good cpu for the job.
I gotta agree with @fredrich_nietze; buy used. You'll save so much money buying used, and could potentially get a lot more performance. If you can't afford X79, go for good-ol LGA 1155 (second generation).
If you buy from EBay, you can get whole systems for real cheap (like $200) with i5/i7's second gen. If you want to install a graphics card, well, you'll almost guaranteed need to upgrade the PSU and do a bit of modding to get it to work, but it is doable (on some pre-built motherboards! Do some research! The Lenovo M81 allows you to upgrade to any PSU).
Look at a skylake I5. You'll get the best gaming chip around, and can use QuickSync in OBS to stream without putting much any load on the CPU. QuickSync basically does the encoding on the iGpu, eliminating any stress that was otherwise on the cores.