Thanks question - I am Sam and I am one of the UK tax experts here on Just Answer.
Your stepson will need to register as an employer (if he has not done so already) and complete a form P46 , which states that this is your only job, but that you do have a pension.
HMRC can then establish that of the £10660 you can earn each year, £3276 will be used up to take into account your state pension, leaving £7384 a year to earn free of tax. As this income from employment of £150 a week will earn you £7800 (on a full tax year) you will just tip over the tax free limit, and need to pay tax of £83.20 per year on the £416 in excess of this threshold. This amounts to £1.60 a week. You are correct there will be no National Insurance, as you are state pension age, but there is just that little bit of tax.
But the receipt of the form P46 will allow HMRC to issue the correct tax code to be used each week against your pay, thereby deducting the right tax.
I have added a link here stepson regarding registering as an employer, if he is not already. https://www.gov.uk/register-employer
But let me know if you need any more information regarding your position of that stepson