I have a computer with a 4-core 2.something Ghz processor and 8Gigs of RAM. It doesn’t have a great video card, just some old thing that allows us to have two screens (Which we don’t even use). We have three USB webcams plugged into it. We’re using OBS as our streaming software. What single purchase can I make to get better performance? Like, should I have a better processor, or do I need more RAM? Would a better graphics card be useful? Bear in mind that only cameras and some overlays would be used. We aren’t streaming games…
I would go with a better processor, that will get you some good mileage.
Another thing to keep in mind is to spread out the webcams a bit if they are newer and have a higher bitrate. I was pulling hair a year or two back trying to figure out why multiple webcams kept bugging out, it turned out the USB hub was saturated and I just had to move a camera to the next port over.
100% better processor probably would be best to do entire platform update tho because guesing you are on an older intel which means upgrades arent really going to be much better.
Processor would be 1st bigg upgrade, in divvying up a/o speeding up the various demands [Webcam inputs, Bkgd / Capture Software, plus Game]. Then memory be next point of interest
This is all dependent, on what mainboard you have, in identifying how much of an upgrade path is available [before bringing up a full update]
I would argue the best single upgrade for streaming would be a proper comfortable chair - not the gaming chair non-sense, mind you - but a true ergonomic chair. You will be seating on that chair a long time and you better be happy and comfortable.
After that, a better lighting setup. Webcams love a bright room and makes the videos cleaner and less grainy with just more light.
You dont seem to need a good video card if it is just transcoding jobs, so a nicer Ryzen CPU should do the performance uplift you need. You may not even need to get the newest motherboard, probably just the nicest CPU you can afford right now on your current motherboard.
Just like many other said get the best possible CPU available for your platform or just get into a new one.
Another good upgrade would be getting a Nvidia GPU with the NVENC encorder to offload the video encoding to the GPU.
Whatever is the cheapest for you to get your hands on is the better option.
I don’t think the OP is referring to web streaming for an audience but just streaming a bunch of webcams.
Yeah, that’s not important at all. We already have good lighting, cameras, etc. In fact, that’s where most of the budget went for this project: aesthetics. My initial thought was to get a newer computer. We have several older (circa 2015-17 era, some even older around 2010). The upgrade path of these are all Intel-based, so really not very good. I think the most power I can hope to achieve is like 3.2Ghz on a 4-core processor.
I think I’ll push for a six-core or better Ryzen series chip, new mb, and ddr4 something or other. Just try to bring into the present day…

six-core or better Ryzen series chip
Depending on the resolution, framerate and bitrate I’d say that a Ryzen 1700 or a 2600 is the best value for your setup. Even some Intel can be good, like the i7 8700 or i7 9700 if you can find one. But, for the AMD machine, you’d need to add a graphics card, to get a video out. Any would do but still the market is crazy right now.