If you are inbound into the USA, your luggage tag will show both flights (on a normal connection of several hours), but the bag is offloaded at the port of entry and sent to the customs baggage return for you to bring it through customs. If your connection time exceeds the maximum allowed by your airline, your bag tag will only show your flight to LAX from LHR. You would get the LAX-HNL tag the next day during check in.

Outbound from the USA, the bag is checked through to the destination, even if you have two or more connecting flight segments. However, if your connection time exceeds the maximum allowed by your airline, they will only check it as far as the international gateway connection city (in your situation, LAX) and force you to get a new bag tag to LHR when you check in for the flight to London.