The only RTR's i can see which include 2.4Ghz within budget are the Ansmann 2wd Mad Rat/Macnum truggy (£99) though this may well prove non child proof, lol. I believe the arguably tougher LRP Twister 2wd buggy is available with 2.4Ghz (according to ebay..), as to whether parts are available to you i don't know...
We have one or two Electrix(ECX) Boost 2wd buggies at our club, these are Made via Horizon Hobby (Losi...) & seem extremely tough, built for beginners, basic but still tuneable. Retail for approx £110 truly RTR, not 2.4Ghz (though 2sets of crystals supplied):
http://www.horizonhobby.co.uk/autoon...ost_buggy.html
As a curve ball, if you want Kyosho, these look kinda cool:
http://www.modelsport.co.uk/kyosho-s...roducts/368739
2.4Ghz, RTR (minus battery/charger), maybe easier for you to get parts for & very eye catching for the beginner.
The decision is made easier if you're not looking to upgrade too much as you'd need something which is either expensive or a bit highly strung for beginners.
There are lots of kits/RTR's out there but many will be ruled out due to either spares issues, durability, cost etc etc..