Thunderbolt Display compatibility with Mac Mini M4

If this is a 27" Thunderbolt Display whose hydra cable has two heads, get an Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter .

If this is a 27" LED Cinema Display whose hydra cable has three heads, get a third-party USB-C (male) to Mini DisplayPort (female) adapter for video. Connect the USB head (e.g., with an Apple USB-C to USB Adapter or a USB-A port on a hub or dock) as well, so that you can control the display's brightness.

If you get the wrong type of adapter, the connectors will physically fit, but you will get no picture, as the adapter won't be delivering the kind of signal the display expects.

With the 27" Thunderbolt Display, sometimes the hydra cable is bad or flaky. Recent versions of macOS are less tolerant of transmission errors than old ones, and will intentionally cut off the picture or reduce resolution if they detect errors. If you run into this issue, you could try running a Thunderbolt 1/2 cable from the TB2 port on the adapter to the TB daisy-chaining port on the monitor, to bypass the permanently-attached cable.

As far as I know, there are no adapters that go from HDMI (computer) to Thunderbolt 1 or 2 (display). Even ones that go from HDMI (computer) to Mini DisplayPort (monitor) are few and far between. Typically, adapters with a mDP plug on one end and a HDMI plug on the other only support going from a mDP source to a HDMI display.

Thanks for the great answer. I had already ordered the Apple adapter for my dad, but was still confused about the mini displayport sign on the connector. Now I could cancel the order and order a cheap mini displayport to USB-C adapter to connect the new Mac Mini M4 to the LED display.