That part is usually called a quick-release clamp or quick-release adapter.

It normally matches a specific quick-release plate design, so replacement clamps are not universally interchangeable. Many brands use their own plate/clamp system, and compatibility is often best within the same brand. Check your tripod/head brand first for compatible replacements.

If you’re buying a new system, an Arca-Swiss compatible clamp is a common choice because it’s widely supported by many accessories such as plates, L-brackets, and replacement lens feet. That said, cross-brand compatibility in Arca-style gear is common but not always perfect.

Also, check how the clamp mounts to the ball head itself: the top of the head may have alignment slots or a specific mounting pattern that the new clamp must match.