The answer is "mare".
The expression "as scarce as hen's teeth" means that it doesn't exist (because they don't). The original meaning of "mare's nest" is something illusory--or doesn't exist.
In addition to the parallel meaning between the two expressions or idioms, there is also a parallel construction, with a possessive in each case: hen's teeth; mare's nest.
This is important in an analogy question, which requires an exact parallel for the answer to be correct (A is to B as X is to Y).
The parallel construction provides the test against which other "possible" answers to this analogy can be tried. There are plenty of things that don't exist, for example, but only one expression or idiom that shares with "hen's teeth" both meaning and the possessive construction.