If it helps in anyway, i've just gone through the K1 (marriage) visa process, and i'm now in the process of gaining work authorization (EAD), becoming a legal permanent resident with intention of gaining US Citizenship 3 years down the line.
From what I understand, the E1 is a nonimmigrant trading visa, enabling non US citizens to trade (and live) within the US. From looking at the visa information, E1 spouses can apply for a work permit and go to school, but children can only do the latter.
The problem, unfortunately, is that you're not classed as a US Citizen (or legal resident) thus neither are your spouse or children. This is going to be the major problem.
Now, if your son can somehow get his hands on an H1B visa, maybe through work sponsorship, that could open up some other opportunities.
Not anyone can gain an US social security number though; you need to prove your either a US Citizen, or show your immigration status and/or work eligibility. I was able to gain one through the K1 visa.
Personally speaking, my first port of call would be to visit your local Government Immigration or work related office in your 'home US' town, before shelling for an immigration lawyer.
For what it's worth, we did our own paperwork for the K1 visa. So far it's cost us around $800, but that was just for the filing fee and various postage. I would hate to think what an immigration lawyer would charge, as all they really would be doing is filling out and submitting the paperwork.
As fernytickles suggested, go look at british-expats.com. While it mainly deals with marriage/family based visas, it may also be worth looking at
www.visajourney.com
I wish you the best of luck with your task !
(please note, I could be wrong in any of the above information, so please do not take it as 'the facts'. Do your own research; I just have an active interest in all things Immigration wise, due to what we've been through !)
Doomie