There are a number of nice third-party gamepads available on Amazon like this one. While the instructions tell you that in order to connect it to a PlayStation, you need to first connect it via USB, there are no instructions on how to connect it to an Android device or Laptop using Bluetooth.
First, you need to put the device into pairing mode. For my device, this worked by pressing both the Home and the Share button for approximately 5 seconds, until the LED bar started to blink.
When the LED bar blinks, you can immediately connect to the device using Bluetooth.
Note that just long-pressing the Home button made the LED bar blink but didn’t work – you need to press both the buttons circled in red.