You can buy a tiny ESP32 microcontroller with built-in WiFi for extremely cheap. (So cheap, in fact, that a friend who bought me some microcontroller stuff as a gift accidentally bought a pack of 3 and thought it was the price of only 1.) A little button switch is fractions of a penny (though you'll likely have to buy a pack of many at a time). Then the code to trigger an HTTP request on a button press is a simple modification of the example HTTP client code that comes with the Arduino IDE.
So... yeah, that's the cheapest route in all likelihood. As in, you can do it all for about 3 to 5 dollars.
My guess is that the friend wants a nice button, like the one in the America Got Talent. Bonus points if it can be relocated in the desk, so perhaps add a battery and a suction cup, and a led and buzzer to confirm.
The button can be put into any case as desired, of course; it's just a switch. Not sure about the buzzer; those can get annoying. But if the HTTP request gets proxied through a little server on the friend's machine, then the server can play a nicer confirmation sound, add notifications, etc. as needed.
But yeah, if they want full wireless, a USB battery pack will do the trick.
So... yeah, that's the cheapest route in all likelihood. As in, you can do it all for about 3 to 5 dollars.