When Ive had to stay near Twickenham either for work (i be a steward there) or watching I stayed in Twickenham itself at the Travelodge - but Id rule that out - and I doubt youd get a room now either
places like Feltham have a travelodge and the St Giles hotel...this is about 5 mins on the train from Twickenham and both are near Feltham train station.
Staines is a bit further away and has a decent Travelodge - but it is a 20 min walk from Staines station
If you are worried about sites, always go with booking.com, hotels.com etc - or the official site of the hotel chain.
Personally Id recommend the St Giles in Feltham...yeego.com reckons you can get a room for £50 that night