Would you please mention the official website of H1 visas lottery.
Well there isn't really a website as such - and the lottery isn't one where anyone can enter and the lucky winners get picked regardless of their qualifications.
For an H1 the process is like this.
1. The company interviews the employee and makes a judgement that it will be worth their time and money on hiring this person.
2. The company applies for labor certification (to justify the need of the role to hire a foreigner). This is LCA. This step costs a few hundred bucks plus lawyers fees.
3. The company submits the LCA and a set of documents explaining how the candidate is a match for their need. The candidate has to be qualified and usually experienced. This step costs the company a couple of thousand bucks plus lawyer fees (around $2k minimum).
4. Step 3 is timed to arrive at USCIS on April 1st (for H1 visas that will start in October at the earliest). USCIS receives all the applications and if they have more applications than they have visas available they then conduct the lottery. This year they received 120k applications in the first week, for 65k visas. So - they held a lottery. Any company that submitted the application would have spent about $3k minimum and got nothing. It is risky for them - so why would they bother when there are plenty of young American kids who want the same jobs.
Honestly an H1 is not realistic for your son. You should not proceed in the hope that you can fix this through the H1 route. As previously explained you could apply for a student visa for your son, but given your new immigrant status it will be very clear that the intention is to adjust status later and that could present issues. Plus the cost of attending US school should be considered. However that is the only way I can see you proceeding.
Then again as you have previously explained you currently have a good job, good benefits and your family is together. I don't know about your quality of life where you are, but consider this. Is the dream of moving to the USA worth it? With all the risk it entails
and all that you will be throwing away
and the likelihood that your family will be split up. I advise you to reflect on that very carefully....