Which Linux Distribution Would Work Best on an Older ASUS X510?

That laptop should be perfectly usable for your needs. I’ve seen similar hardware run a minimal installation with XFCE and a basic browser surprisingly well, including email, ordinary browsing, streaming video, and even VPN-connected office work. Keep expectations realistic with modern websites, since browsers can consume a lot of memory, but 4 GB should be workable if you avoid having too many tabs open.

Linux Mint with the Cinnamon desktop can also work on this kind of machine, especially if you prefer a more familiar and polished interface. XFCE or MATE would leave more memory available, though, so I’d choose one of those if browsing starts to feel sluggish. The integrated Radeon R7 graphics should be supported well enough for normal desktop use and video playback.

A Linux distribution using XFCE should be a good starting point. It’s fairly lightweight while still feeling like a normal desktop, and it should use roughly 1 GB of RAM when idle. If you want something lighter, MX Linux with Fluxbox is another option, though Fluxbox feels pretty dated. antiX is even more resource-friendly and remains actively updated, but its interface is less polished and aimed at much older hardware.