Is an elliptical orbit around a black hole, or an orbit consisting of three bodies (including the black hole) possible? If this is possible, would it then lead to an object undergoing varying time dilation (predictable variation with elliptical and random variation with three body)?
Elliptical Orbit

Yes, an elliptical orbit around a black hole is possible.  Provided the orbit doesn't cross too near the event horizon, the orbital mechanics are the same as orbiting any massive object.  Gravitational time dilation is a function of distance only, so the time dilation experienced at any point in the orbit should be easily calculated and presumably predicted ahead of time or calculated for past times.

Three Body Orbit

When you mention an orbit consisting of three bodies, do you mean the black hole, another massive object, and a much smaller object?  Or do you mean three massive bodies?


If the orbit consists of the black hole, another massive object, and a smaller body, the system should be predictable as a restricted three body system.  This is made even easier if the smaller body finds its way to a Lagrange point, or resides in a stable repeating orbit.  There are limits to the predictability of the restricted three body cases, which differ based on requisite constraints (e.g. circular vs. elliptical orbit for the smaller body).  If the constraints are not met, or near their limits (not hard limits), the predictability of the orbit will suffer and thus the predictability of the time dilation will suffer as well.  Constraints for the elliptical restricted three body problem include the mass of the small body being "negligible" relative to the two larger bodies, and the ability to approximate the co-orbit of the larger bodies as Keplerian between point masses.  Thus if the mass of the smaller body is some appreciable fraction of either larger body, or if the larger bodies orbit very closely to one another, this may not hold.  The circular restricted three body problem adds the constraint that the co-orbit between the larger bodies must be circular, so if it is near-circular that condition may or may not be said to be met.  Whether constraints are met would be most easily judged by setting a time over which predictability of the system may be maintained.  So if the goal is to be able to predict the system for 1000 years, and the smaller mass is small enough that the system is predictable under the restricted three body problem, the system can be said to meet the constraints for those purposes.
If the orbit consists of the black hole and two other massive bodies, the system will not be perfectly predictable (this may not be totally true, apparently solutions exist in the form of convergent power series though they may not be practical due to their size).  If this is the case, time dilation will not be predictable, but will be calculable for the present and past (provided there is position data from the past).