If you fly Rome to LAX to HNL, you will go through both immigration and customs in LAX. The immigration line can be long or short, and how long it takes cannot be predicted. It depends on how many agents are on duty, and how many other international flights arrived at your terminal at the same time (and at LAX, there are always going to be other international arrivals). You then pick up all your checked bags, then go through customs. In the US, there is only one customs line; you can certainly be stopped for additional screening (although it's only happened to me once, returning from Hong Kong via San Francisco). Just outside the customs door, there is a baggage recheck desk, where you deposit your checked bags so they'll be put on your next flight. You then make your way to your next flight, going through security on the way (since you've left the "sterile area").
All of this takes time, particularly if you have to change terminals at LAX. I agree that the Mobile Passport app can save oodles of time (at least it has at JFK and Newark the three times I've used it), but I would still allow several hours for all of the "formalities." One advantage you have is that there are multiple flights per day on all the airlines between LAX and HNL, so if you do miss your connection due to a late arrival at LAX, you should be able to get a flight home later that day, rather than having to wait until the next day. As long as you're all on one ticket, and as long as the missed flight wasn't your fault (i.e., you didn't miss the flight because you were busy shopping), they should put you on the next available connection at no extra charge.