There are 3 decent places (as another parent struggling to find decent toys) - If you get a taxi from the port to M&S (passing and avoiding the ridiculously expensive and touristy 'Old Town' - next door is a good little toy shop, similar to a Disney Store type shop in the UK. From there ask where McDonalds is, it's just a short walk - on the corner facing McDonalds (McDonalds is on a corner too) is another shop, half tacky tourist souveniers and the other half general toys and games. If you need some more, grab a taxi to the above mentioned Lidl/Spar. The Spar is very big and has a toy section. None of these 3 have huge toy selections, but together way better than anything locally.
I forgot to mention, the Taxi driver may ask which Lidl/Spar you mean - they are on the Lindos road out of Rhodes town, and either side of what is a pretty busy road. Getting a taxi back may take 5-10 minutes to flag one down, but you will get one.