I would like to see more literary analysis of the early texts, analyzing the symbols, mythology, narratives and motifs less so from the lens of historical authenticity which prevails so much today. This will, in fact, shed more light on history, as myth and symbol can end up indirectly saying a lot more about time and place than many other more direct mediums. It would also be nice to see more analysis of the similarities, differences, and potential motivating forces behind the differences that are found in the narratives and symbols across Buddhist schools and textual recensions. E.g. why does one text say this, and another text say that? Situating these symbols in the larger Indian context would be very helpful.
We could really use people who study the Chinese Āgama material on its own terms. That is, rather than using it as a sound board for the authenticity of Pāli material, we need people assessing the doctrines and ideas found in the Chinese translations themselves.
Ideally this would be informed also by studying the early Abhidharma material of the Sarvāstivādins, Dharmaguptakas, etc. That is, getting an idea of how the different schools and texts understood Buddhist concepts. This will help understand some of the translation choices, school affiliation, etc. and provide a more complete picture of the early corpus, rather than trying to decipher it without understanding the other Buddhist material that surrounded and informed the texts themselves and the translations.
And on the topic of Abhidharma and early schools, it seems it would be helpful to have more research on the development of common sectarian Buddhist material post-Ashoka but still within the first ~500 years. For example, looking closer at the origins, evolution, and relationship to the Abhidharma analyses and how it fits into the sutta corpus of the schools. Apadānas, jātakas, and commentarial stories being compared and traced down. Later vinaya narratives and their evolution. Things like this that give an idea of what the later ‘middle Buddhists’ were thinking, how they related to their texts, what ideas mattered to them, etc. And ideally all of this in a way that does not favor the Pāli corpus automatically but rather uses it as one of many equal tools.
Some of this will run into inter-religious territory, and here I think we need to get a better understanding of Jain influence on Buddhism. There is clear and obvious influence both early on in the suttas and continuing into many centuries later. We need more study of the Jain ideas in earlier periods (incl. after Mahāvīra; rather, the broad timespan when they were interacting with early-middle Buddhism). We need study of Buddhist responses to Jainism and other sramana movements like the Ājīvikas. It would also be good to get a better understanding of Brahmanical-influenced philosophical systems and mythology, like early parts of the Epics and emerging later Upanisads or Sankhya texts that interacted with Buddhism. These had a clear impact on Buddhist terminology and later development, so understanding how and where some of this began earlier on will give a better understanding of the evolution.
There is also the impact of non-Indian ideas, especially say the Greek influence. As far as I know there is a gap in understanding specifics of how Greek ideas influenced the Abhidharma material, and how trade and technology played into the development of Buddhism. This includes things like the influence writing, politics, geography and financing. As an aside, and another big topic: how did Buddhists organize themselves? How were monastic communities organized and how were buildings designed and run from the mentions in the suttas and vinaya into later periods?
Just some ideas.