Thompson has held a number of senior marketing roles at O2 and its parent, Telefonica, over the last seven years.
Prior to becoming Telefonica's innovation chief, he was European director of marketing innovation for O2 and Telefonica.
He founded Giffgaff in 2009, becoming the budget network's early-stage chief marketing officer.
Thompson will be responsible for Thunderhead's strategic brand direction and will oversee all marketing activity.
He said: "The marketing landscape is changing faster than ever before.
"Thunderhead has anticipated this by developing a completely unique and ground-breaking approach to customer engagement, with a solution that is purpose-built for a 21st century CMO. It will be a great privilege to help the team continue their launch of this powerful capability to the global CMO community."
A spokesman for Telefonica said there were no plans to replace Thompson at present, saying: "There are currently no plans to replace his role directly, but innovation and digital services continue to be an essential part of Telefónica with Ian Small leading the communication services and product innovation team reporting into our chief commercial digital officer, Eduardo Navarro."