I see two relevant possibilities for a Brazilian citizen to bring goods bought outside the country in this specific case:
- Products for personal use are duty free as long as they where needed during the trip and are strictly for personal use (that is, not to be sold or being given as gifts). This official brazilian government page gives as an example: a photo camera, a watch and a cell phone as long as they show signs of being used during the trip to a foreign country.
- Besides that, Brazilian citizens can bring up to US$1,000 (one thousand american dollars) duty free when returning to Brazil, by air or sea. So it might be an option for some (perhaps all?) items that your friend intends to buy in the UK.
About your second question, how can your friend prove (s)he had something before leaving Brazil, the offically suggested way is to have a reputable document (such as a Brazilian invoice) of the product which is given when the product is bought. Showing history of calls (or something like that) is not guaranteed to be accepted (even though authorities have some discretion when deciding what will be considered a personal good). Anecdotally, people where buying stuff like laptops and loading the hard drive with an image of an operational system with lots of (old) files, as a way to "prove" it was previously owned.
There was a time when you could show up in the Receita Federal (the government service that is responsible for taxes, sort of like IRS in the USA) and present your goods before departing. They would give you a document stating that you had such items before departing, but this is not happening anymore.
Just for the sake of completeness, here is a link for the official "traveler guide" for Brazilian citizens. It states all the currently valid rules and exemptions one might use. You friend better read it carefully so he knows all the details.