Big data stream mobile computing is proposed as a paradigm that relies on the convergence of broadband Internet mobile networking and real-time mobile cloud computing. It aims at fostering the rise of novel self-configuring integrated computing-communication platforms for enabling in real time the offloading and processing of big data streams acquired by resource-limited mobile/wireless devices. This position article formalizes this paradigm, discusses its most significant application opportunities, and outlines the major challenges in performing real-time energy-efficient management of the distributed resources available at both mobile devices and Internet-connected data centers. The performance analysis of a small-scale prototype is also included in order to provide insight into the energy vs. performance tradeoff that is achievable through the optimized design of the resource management modules. Performance comparisons with some state-of-the-art resource managers corroborate the discussion. Hints for future research directions conclude the article.