Don't worry about your child being in business class - business people tend not to fly on Saturdays anyway. My son's been in Club World on BA in the past and no-one took a blind bit of notice of him.
Hmmm, as for any potential upgrade that may be a bit more tricky. You can only but ask but they may well refuse. You may not get an upgrade for free either.
The only other airlines I can think of that fly to Maui from LAX apart from United are Aloha, Hawaiian and American.
What I would suggest you do ahead of time is to be fully conversant with where you need to go once you have landed at LAX. The last thing you need is to try to work out where the United domestic terminal (Hawaii is classed as a domestic route) is when you are there.
Also bear in mind that all departures are located on the upper level; you will arrive on the lower level.
Hope as well you've left plenty of time before catching the flight to Maui - you would need around a 2 hour transfer gap to be safe methinks.
Also ensure that all the forms you need for immigration and customs are completed well ahead of time - you do NOT want to stand in line at LAX completing the things. You will need to complete one customs form and three visa waiver forms (a parent will need to complete and sign one for your son).
You will also need to complete a form on the plane to Hawaii (this is collected on the plane before arriving into Maui). Its a combination of both a customs form and survey. This form is compulsory, not optional.
Love Maui loads, have a fab time!!.