Since many difference equations arise from discretization of differential equations, one source of open problems is to begin with a question about ODEs (or even PDEs). One possible direction, from the control-theoretic area might be to look at time-sampled systems.
My interest lies in the modeling of systems described by difference and algebraic equations. More specifically, how can one, given a set of discrete time vector functions, create a system that has these functions as solutions. This problem has been solved using algebraic methods (you can check my papers if you want), and also through behavioral theory, e.g. Willems, Zerz e.t.c. One interesting question is when the constructed system is the most powerful, i.e. it has the given solutions, but no more than these. Willems tackles this subject, and I also happen to have submited a paper that has to do with this.
I also believe the ideas of Thomas I. Seidman above sound really interesting.
Journal of Difference Equatiions and Applications has an entire section on the subject
Article Open Problems and conjectures
See also
Conference Paper Advances in Periodic Difference Equations with Open Problems
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