What is the name of a device used to detect noisy appliances?

For first-pass broadband EMI fox-hunting, I use an old pocket-sized transistor AM radio with a ferrite loopstick antenna. You can get them as electronic kits, which makes for a nice educational construction project for the kids. When constructing such a kit, I left the loopstick antenna exposed as an LF/MF EM sniffer probe.

I tune between stations, so that I only hear hiss when away from anything powered. The old AM radio will start buzzing loudly when I get roughly on the order of a meter or so from the offending (LED lamp, USB wall wart, appliance, etc.) EMI emitter.

This is several orders of magnitude cheaper than a spectrum analyzer, and vastly more portable as well, for finding what to try unplugging. Then recheck with your HF/VHF radio to see if the noise floor looks better.

If I'm shopping for such a detection device, what are the keywords I should be using? What is the name of the device I am looking for?

Now, I'm interepreting this as

I'm looking for sources of radio-frequency noise, and I need to find them, for reasons that are professional and not limited to a specific interference ocurrence.

The device that you want to use for that purpose is a Spectrum Analyzer. It is a measurement device that shows at which frequencies there is power "in the air". It typically works over a large range of frequencies, e.g. from 800 kHz up to 8 GHz.

You'll also want a set of antennas that work on a very wide band, so you don't have to switch antennas for every single band you test.

Just like the me measurement device, if you want these power readings to be physically meaningful, your antennas need to be calibrated, if you want the output to actually represent the power that is in the air. This is kind of important, considering "off by a factor of 100" not unusual for antenna properties for the same antenna at different frequencies.

Such (entry-level) devices look like this

or this

You'll pretty quickly notice the price of these things is in the region of "car" if you have modest requirements. (There's literally no limit upwards – at some point you get into the range where Keysight will offer you a custom solution, but then you're in the ballpark of megaeuros.) The price very much scales with the range of frequencies that can be observed, so in the interest of your savings account, you will need to first understand what you're trying to measure ("noise" isn't really telling us anything), and then define requirements for your device from that.

There are things on amazon that are scams, but try to take money from people worried about "noise" with no understanding of what that is. Don't buy them, they are a scam. I've covered that before.

Could I make my own?

Technically: yes. A spectrum analyzer is a technical device, and given theoretically unlimited resources, anything in existence is possible to engineer.

How tough would that be?

Since you had to ask what kind of device you needed, and it's (and that's OK) not clear you fully understand what you're measuring there: In all honesty, very tough.

An electrical engineering degree with a focus on RF electronics, plus signal processing and estimation knowledge would be a plus.

It takes decades of experience and large engineering teams to produce sensible spectrum analyzers.

You could try to implement one based on an SDR, but the problem will be making it reliable enough so that a calibration would still mean something the next time you turn it on. Also, there's analog reasons that SDRs are available cheaper than spectrum analyzers, and it's that the filters and dynamic ranges necessary for a measurement device are simply way more expensive to buy or build than what SDRs come with.