I think having a fan in the boardroom is a step in the right direction. This person should come from a pool of season ticket holders that have held their ticket for at least 5 years. They should also have to be voted for by other spurs members and should be voted on every season or two. (the same person can be selected again if they are doing a good job). A simple CV and video on the website would be enough.
What would their role be?
Well I agree that they will not and probably should not have any direct power in decisions over key issues, such as finance, players, managers etc. They wouldn't even need to be at every board meeting, however, they should be invited to specific meetings where they would be able to bridge the huge divide between the extremely aloof board and the supporters. They should be expected to express concerns by the fans over certain areas, so those views were at least directly heard and taken into consideration. They should also be able to share what was discussed during these meetings and help the fans, which are usually left in the dark, with some important information regarding the club. This would provide a needed layer of confidentiality, the board would not want every discussion made public. This person could at least be given the opportunity to ask the questions which the media too often neglect to ask. The board can decide how to answer these questions, but they should be expected to give some sort of response.
It won't solve all the problems we have, but it would be worth trying.