This is not one of the usual weeds that get into lawns - I suspect it's an escapee from a border, and may be one of the purple leaved sedums. These do spread, usually in clumps, but it's more likely to spread underground and pop up further away from being cut regularly, as it's invaded the lawn.
You don't say what weedkiller your neighbour has been using, but if it's one of the combined lawn preparations (usually feed and weed or feed, weed and mosskiller) this won't work. These granular preparations are relatively ineffective as weedkillers, so if he's not tried Verdone extra (now sold as Weedol lawn weedkiller concentrate), he should try that, preferably a week after cutting the lawn,and without cutting again for another few days. It is quite strong and one application is all that's recommended for the growing season. If he wants to feed the lawn as well, he should buy a lawn feed in a single formulation with no weedkiller content. The only other alternative is to dig out clumps wherever they occur, fill in with fresh topsoil and seed over.
If the lawn is mostly weed with very little grass (say 30 to 40% grass,the rest weed), then it's better to take up the lawn, remove all the weeds by the roots and relay new turf.