So then, the question:
"Would my son have to declare the shares as his income, for instance..."
No. They are not income. If he sells them he may be liable to CGT depending on the value of them when he acquired them and the value when he sold them (assuming the gain took him beyond the CGT annual allowance).
"...or only the feeble twice-yearly payouts ?"
In the current financial year £5,000 of dividends can be received free of tax. On receipts greater than that you pay tax at either 7.5%, 32.5% or 38.5% depending on whether you pay tax at the basic rate, higher rate or additional rate.