You might want to start with an 8 GB memory stick by SanDisk or Kingston (established memory firms) but for 8 GB, suggest you should not spend more than say a tenner (inc delivery!). Firms like Play.com and Amazon should have one or other brand. Other firms might be found using Google 'Shopping'.
When I download shows using iPlayer it is common for half-hour shows to fill 300 MB and hour-long shows about 600 MB. How much space you need for an hour will depend on whether your box is recording in High Definition or Standard Definition (if you have both options) and how much compression is used. I've suggested 8 GB as you should be able to see how much each item takes and how many hours capacity 8 GB gives you (without it being filled up first time you record a show, which might happen if you went for 1 GB or 2 GB and saved a long film, say).
Something you might have/consider is an external hard drive. There are firms selling at modest cost on Ebay (Daily Saver deals)... eg 250 GB for about 30 quid.
Now I cannot guarantee it would work with your TV recording box, but if it does, and has no problem even when you have several dozen items recorded, it would mean you could keep lots of episodes of a show, or record some films for viewing on wet November nights!
If it doesn't work with your TV box, at that price it is not a massive 'bank breaking' purchase, and you could use it for backing up photos and music off your computer instead.
Just nosy, but which makes/models of TV have you bought, please?