That's not easy! The easiest and most cost-effective are certainly not going to be the same.

You could get a 285 bus to Kingston station, you'd then need another, or a taxi, to Surbiton. That would be cheapest but going to take over an hour.

Tube into London, then mainline out again-long journey.

Taxi - going to be easily £40 each way I'd have thought, and from LHR you'd need a driver who will cross the Thames (some are a bit arsey about that.)

Might almost be best to hire a car! Theres a few offers going on at the moment.

Alternative - have you thought about staying at a Heathrow hotel? The CP is excellent and upgrades plats on points nights, as you'd still have to get from HI Kingston to Twickenham and back, gonna add up to a lot of cost and or bother.