A faster Athlon II X2 like the 260 can probably manage a "passable" job with many games. If you are building a system yourself, however, another $20 or so (here in the USA) buys a Phenom II X3 740BE, a significantly better chip (although you'll also need to buy a $20-$30 HSF for it). If you are NOT building this system yourself, carefully check for what chip it is. Some low-end machines are being offered with the Athlon II X2 240e and 210e. These are the slower, low power versions, which are fine for Internet surfing and email, but would be unsatisfactory probably even for Flash games.
Edit: The 740BE also has a reputation for unlocking easily to a X4; mine did just by changing one setting in my ASRock board's BIOS.