I have just been thinking about it for a while and would like to see if there is a way to do this problem.
We have an insulated cup with mass $m_c$ and specific heat $s_c$. The cup is at temperature $t_c$. We put a liquid with mass $m_l$ and specific heat $s_l$ in the insulated cup. The liquid is at temperature $t_l$. Assume the freezing/boiling point of the liquid is extreme enough so that it does not undergo any phase change when in the cup. Also, assume that these two objects are the only things affecting the temperatures of the liquid and the cup. (that is, ignore the atmospheric temperature)
We can easily find the equilibrium temperature by using the familiar $Q=mc \triangle t$ and summing, setting the sum equal to zero.
Is there a way we can find the equilibrium time? Of course, since we are given definite masses, specific heats, and temperatures for all the objects in question (in the form of variables) it seems that the answer should be expressed as a function of all those (with the addition of possible some unnamed constants). Is this known? If not, how could you find it?