What lightweight live OS can I use to test a laptop with no boot drive?

You don’t necessarily need a specialized lightweight distribution. A current Ubuntu or Debian live USB should boot on a 2014–2016 laptop and gives you plenty of hardware-detection tools. Try booting it in test mode, check whether the drive appears, and run memory and hardware diagnostics. That hardware isn’t especially old, so most mainstream distributions should run fine.

Grml is a good fit because it includes many troubleshooting and hardware-checking tools. Puppy Linux or antiX would also work if you want a lightweight desktop environment. You can boot any of them from a USB drive without an internal disk, then use terminal tools to inspect the CPU, memory, graphics, networking, and attached drives.

If the goal is specifically hardware diagnostics, a bootable WinPE toolkit such as Sergei Strelec’s environment can be useful. It includes utilities like CrystalDiskInfo for checking the drive’s health, along with other diagnostic programs. Create a bootable USB, select it from the BIOS or UEFI boot menu, and check whether the drive is detected. If the drive reports poor health or is missing entirely, replacing it with an SSD is probably the next step.

If the hard drive has actually failed, the operating system choice won’t make much difference until you replace it. A basic SSD would be a major improvement and you could install nearly any suitable Linux distribution afterward. For now, though, a live USB is enough to confirm that the laptop, memory, display, keyboard, and other components are functioning.