Generally, it depends on the character of the property before the state took possession of it as unclaimed property. If a company remitted dividends to the state because the company could not locate the beneficiary, then those dividends would have been taxable income to the beneficiary. When the beneficiary claimed the "income" from the state, it is probably taxable when received. 

In a different case, if a bank remitted a dormant checking account balance to the state, then that cash would not have been taxable income to the owner and would not be taxable when the owner claimed the "property" from the state. 

In your case, it sounds like the stock would have been "inherited property" if it had passed from the owner to your taxpayer. That would not be taxable income to your client. So absent a state statute that says unclaimed property recovered is income, it doesn't sound like taxable income.