What are some promising areas of research in digital signal processing?
Choosing a novel topic is a bold and wise move. Your question requires a bit of a bet, yet it is interesting to me, as I have not thought about this for a while. I will first drop two dual thoughts (they could be updated):

shed light on black-boxes: the present successes of machine learning and deep networks shall not make us forger that signals are our (DSP community) sensing door to a globally unknown world. Truly understanding what we can understand with data and models remains at our core, and we can be much better at that (to make it more interpretable, robust, model-based, explainable, reliable etc.)
save electrons in devices and computations: sensing and computing are becoming pervasive (IoT, etc.), and this requires materials (from the extraction of rare-earth elements for sensors and computing devices) and resources (form energy-harvesting sensing to bit-wise or qubit-related fast algorithms). Saving power (efficient filters, compression, fast algorithms, etc.) are in our DNA.

DSP is a blended field of knowledge, where people of mixed origins (computing, acquisition, information theory, physics, optimization, computer vision, statistics) meet. I hope that we DSP people can innovate here.
Additional stuff, as they come:

The Future of Signal Processing: conference program (2017)