Which Linux distro should I put on a ten year old laptop?

Do not overthink it. Mint with Xfce, Xubuntu, or Debian with Xfce all run comfortably on 4 GB. Boot a live USB and test Wi-Fi, sleep, the webcam and the trackpad before you install anything. If they work in the live session they will work after installing.

Whichever you choose, turn on automatic security updates. That covers your help desk concern. Keep a spare installer stick around for the day somebody breaks it.

@Raul M. the live session saw Wi-Fi, sleep and the webcam straight away. Trackpad gestures need tweaking but nothing alarming. Xfce it is, probably with Mint as the base. I will report back if the family finds something I missed.

Arch. Rolling release means you always have the newest kernel, which is what old hardware needs to get the best performance out of it. Setup takes an evening. Pair it with a minimal window manager and it will fly on 4 GB.

Chromebook user here, so take this with salt. Is Google's ChromeOS Flex worth a look? It is meant for older PCs and would give your family something close to a Chromebook with automatic updates. It only runs on models Google has certified, so check their current list for your exact laptop first, and whether it is still supported.

@Tyler B. Arch is a fine hobby, but it is not what you hand to a family who wants no help desk. A newer kernel does not make a 2016 laptop faster, and a rolling release means more changes landing on you, not fewer. Pick the boring option.

Part number note: if the Wi-Fi card is Intel you are fine. Broadcom cards are the usual headache and may need a proprietary driver from the installer's additional drivers screen. Check before you pick anything: lspci -nnk | grep -iA3 net tells you the chip and which driver the kernel loaded.

Try the accessibility settings in the live session too, even if nobody in your house needs them today. In my experience screen reader behaviour, text scaling and pointer options differ a lot between desktops, and lighter desktops can have fewer of them. If someone relies on Orca, test it with real tasks before you commit.

@Owen P. it is an Intel card, and the live session loaded the driver without asking. Wi-Fi was the thing I was most worried about, so thank you for the command.