In Web search, users may remain unsatisfied for several reasons: the search engine may not be effective enough or the query might not reflect their intent. Years of research focused on providing the best user experience for the data available to the search engine. However, little has been done to address the cases in which relevant content for the specific user need has not been posted on the Web yet. One obvious solution is to directly ask other users to generate the missing content using Community Question Answering services such as Yahoo Answers or Baidu Zhidao. However, formulating a full-fledged question after having issued a query requires some effort. Some previous work proposed to automatically generate natural language questions from a given query, but not for scenarios in which a searcher is presented with a list of questions to choose from. We propose here to generate synthetic questions that can actually be clicked by the searcher so as to be directly posted as questions on a Community Question Answering service. This imposes new constraints, as questions will be actually shown to searchers, who will not appreciate an awkward style or redundancy. To this end, we introduce a learning-based approach that improves not only the relevance of the suggested questions to the original query, but also their grammatical correctness. In addition, since queries are often underspecified and ambiguous, we put a special emphasis on increasing the diversity of suggestions via a novel diversification mechanism. We conducted several experiments to evaluate our approach by comparing it to prior work. The experiments show that our algorithm improves question quality by 14% over prior work and that adding diversification reduced redundancy by 55%.