As far as I know LAX isn't a particularly cheap place to get YUP fares out of. SFO-MCO may be your best bet but try and avoid the overnight flights if at all possible. I could sleep standing up next to a speaker in a nightclub if I wanted to but found those F seats on the 757 really tough!

http://www.farecompare.com/search/yu...rue&carrier=AA
http://www.farecompare.com/search/yu...rue&carrier=AA
http://www.farecompare.com/search/yu...rue&carrier=AA
and SFO if you can get a flight up there...
http://www.farecompare.com/search/yu...rue&carrier=AA

If you want to go "hardcore" then you could fly LHR-BOS on BA in J, then BOS-SFO-LAX (do business) then LAX-HNL-LAX-SFO-BOS, then BOS-LHR on BA in J.

The AA fares are pretty reasonable out of BOS (The wife and I paid $1700 BOS-HNL-BOS a few months back) and you'll earn a boatload of tier points. 240 for the BA legs, plus 840 for the AA legs, presuming you'll have the four BA sectors to get to Silver of course Your company may wonder why on a trip to LAX that you'll only want a ticket to Boston...This does incur extra hotel costs and what not but you said you wanted cheap options