Until recently, I had no need for a mobile phone. I had a phone on my desk. It no longer functions, our telecom provider removed the phone option when they "upgraded" their service and router, as "nobody uses these any more". So what do I look for when buying a mobile phone. Obviously, I want it to be able to run BOINC, but frankly, I have very little use for it as a telephone, I use my computers normally. I see there are Android phones and ios ones, either better? Other options? What do I want?

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Wave upon wave of demented avengers march cheerfully out of obscurity into the dream.

Why, if you have very little use for a phone to be used as a phone, would you buy one?

Running BOINC on it isn't such a good idea anyway, because if not cooled correctly the battery will bloat and break the phone.

I have very little use, that is not the same as no use. I have a hospital appointment I need to change, and they have quoted a phone number - current issue anyway. My sister in England also deems it necessary.

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Wave upon wave of demented avengers march cheerfully out of obscurity into the dream.

My phone company (British Telecom, using shared infrastructure provided by Openreach) did the same thing to me 18 months ago.

But the BT-supplied router I already had provided a compatible phone socket, ready for use. I did a bespoke modification to my 30 year old house wiring, and plugged in the matching patch cable (about £2), and I have a working fixed phone in every room again. I find that's preferable to carrying a mobile (or even a cordless handset) from room to room.

The drawback is that even the fixed phones lose connection if there's a power cut or the router loses connection. So far, I've been able to use a mobile to ring up BT every time it happens, but in a wide-area outage, even that might not work.

You might be able to get a VOIP router (that's what they're called), either from the phone company or independently. But check with the phone company that they can transfer your number to a VOIP service provider.

There are also GSM desktop phones. They look like old fixed phones but are operating like a mobile phone. They have an antenna and a battery so they can be operated also during power cuts. Normally you keep them connected to the power line.

Personally I think the Samsung Galaxy A72 were the best phones. You can still find them online. Upgrade the os to android 11 or 12, enable power saving (lower performance) and battery protect (charge to 85% & slow charge only), then tell boinc to run 6 out of 8 threads.

The snapdragon inside the A72 has 2P and 6E cores. That's 2E cores more than most other phones.

My recommendation is to keep the phone in a cool place. Like all phones, the battery swells up when too hot.

I set mine to 36C max, which is already too high if you ask me.
But in winter time (below 60F ambient) you can run boinc at full speed (disable power saving).

Last phone shopping I settled in on an 2x LG ThinQ 8's used on eBay with slightly cracked screens, one could be hidden by a cover. $45 ea.
6GB RAM. 256GB storage, 8 cores, nice GPU.

Because of water tight seals, all (not seen a easily replicable battery in a phone for the last 4 years) of these phones have batteries that are hard to replace and best to protect the battery above any other consideration.
There is risk here. My Moto G4 battery started bulging and died, BOINC use is at least partially implicated. At least it was a cheap, used phone (with potential battery issues upon purchase?).
An intense Android game can use a full 8 cores and the GPU.
The phone gets hot, but human hands firmly pressed on the case, and blood circulation, are a phone cooling system.
Running BOINC while the phone is alone needs another cooling system if you wish to use all cores.

For simple BOINC use:
Make sure battery saver is at max.
Install a battery temp monitor app. (Simple System Monitor is my choice)
Consider only using 2 main cores of the 8 for BOINC; increase the core count until you break 36C battery temp.
I've read 40C is safe upper limit but after losing the Moto G4 I'd rather keep it cooler. (Can't wait for the new Silicon/Lithium batteries for commercial use)
There are phone cooling pads and you'll need to remove it's protective case at night to maximize cooling and make sure the phone makes a good thermal connection.