As likely as it is that an adult film actress or a Playboy model would find an attorney willing to sue the President of the United States. No such thing as bad publicity, just publicity.
There are plenty of solicitors who would not be bothered that the Duke of Edinburgh is the Defendant in an action - they may think their client is in the right and believe everyone is entitled to legal representation (which is why even the worst criminals still get someone to represent them). Maybe they are not monarchists/have no feelings one way or the other about the monarchy. Maybe they just relish the opportunity to thumb their nose at the establishment.
Diana had no difficulty in finding a lawyer who was prepared to negotiate a divorce on her behalf. When she sued one of the tabloids for publishing photos of her in her gym, the barrister retained by the paper (Geoffrey Robertson QC) was said to be looking forward to the opportunity to cross examine her about her own collusion with the media until the paper caved. Mohammed Fayed had no difficulty finding a barrister who would run his arguments at the inquest into Diana's death.