Though I don't know the official name, I have for bizarre reasons known only to myself, kept a list of various references to that device, which has included;
little swirly.
Mr Spinny.
beachball
Spinning beachball.
swirling beachball
Spinning wheels
Spinning BeachBall Of Death (Dr Smoke's morbid take on it)
progress wheel
spinning circle
spinning colored icon
spinning colour wheel
processing wheel that never ends
red rolling ball (that's what they said)
rainbow wheel
endless spinball
the spinning coloured mouse curser
agonizing rainbow spinning wheel (and the poster also said)
hypno-coin (and)
prismatic sphere of perpetual motion (and)
spectral spinner
the spinning pizza wheel
spinning colored disc
beachballing (Verb)
rolling rainbow
little swirley load thing (beautiful)
the loading spinner
spinning disk
swirling rainbow mode
rainbow swirl thing
eternal ball