For a 13 y/o, getting a phone like a BlackBerry that doesn't have a lot of app selection could turn out to be an advantage, as you may want him to be contactable by call or message, but not constantly having his head in a Fifa football app.
One option may be to buy the phone outright on PAYG (the BB 9320 is a good pick at around £130), and then get a 12 month SIM Only tariff for around £10 a month which will allow ample talk, text and data allowance. You can then review the arrangement after a year to see how it's working.
Last time I checked, O2 and Voda had some good deals on 9320's coupled with a SIMO tariff.
Vodafone:
£130 for the phone, and then they have a great SIMO tariff.
£10.50 for 300 minutes, unlimited text, 500MB data, BlackBerry services. If you buy online, they'll also throw in unlimited calls to other Vodafone numbers, and 750 MB BT wifi. This is also a 12 month contract.
O2:
£130 for the handset, or you can get a refurbished one for £100
£10.50 for 100 min + 500 extra O2 to O2 minutes, unlimited text, 100MB data, BlackBerry services.
With O2, the 100MB data allowance is throttled - so if you went over 100MB, you could still access the internet without extra charges - you'd just have a slower speed. You would also receive a text message advising the data was being throttled once you went over the 100MB.
If you do go down the BlackBerry route, make sure the data element of the tariff plan included "BlackBerry services" which ensures the email and BlackBerry Messanger (BBM) features of the phone will work.
As for adult content control, this can be done at a network level by adjusting the account profile settings via the operator website. In other words, you set up an online account on the O2 or Voda (or whoever) website and you can then put in place adult content restrictions.
Hope that helps!