The cheapest child return is:
Super Off Peak Return £36.50
(to London Terminals)
However this will not get you to Waterloo in time for the Challenge, therefore a combination of tickets will be required.
An Off Peak Return would get you there earlier, but still not early enough.
Therefore, to answer your question regarding the cheapest child return, the price is a rather shocking £113. You require a Travelcard to do the Challenge (and also bear in mind a return to London Terminals would only get you to Waterloo if you changed at Exeter or Reading for a SWT service!), you can get an inclusive Travelcard for only 50p more, but I am sure we can beat £113.50 by suggesting a combination of tickets (aka "splitting").
No-one has yet devised a program that can calculate the cheapest combination of tickets, so any replies below (I'll leave it to people who are based around that area) are not guaranteed to be, as you request, the "cheapest".
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What time do you need to be there?
The recommended arrival time for the
Challenge is 1030 at Waterloo (if arriving into Paddington I'd suggest arriving into PAD by 1000).