A provision of Testator's New York State Will creates a Trust naming X
as beneficiary of a trust; Trustee is to disburse 1/5 of the trust's
value at Settlor's death to X in each of the next 5 years. (E.g. if
the value of the corpus is $100K when Settlor dies, X is to receive
$20K/year for 5 years.)
   Y is the contingent beneficiary if X dies prior to the expiration
of the 5-year period.
   Trustee resigns, in which case the Trust states that "a majority
of the Beneficiaries shall have the power to appoint a successor
trustee."  X and Y, who are equally competent, can not agree on a
successor Trustee.
   Applying New York State law, does X have a superior interest in
naming a successor trustee, either because X's interest is vested and
Y's is merely contingent upon the death of X, or for any other reason?
If not, what factors would a Court find relevant in determining a
successor trustee?
   Citations, please.

You need legal advice - consult an attorney who specializes in trusts.
 If the two of you cannot agree your only recourse is to sue and have
the court overrule the provisions of the trust.  But why would a court
do that?  If the trust is clear on its provisions why would they
interfere?  I don't think that because the trust was poorly drawn up
would be valid reason, but you never know.  I guess the other person's
strategy is to wait you out and collect the whole amount.  But that
stategy could backfire, they could die first and end up with nothing.

If this doesn't involve a family feud - why not come to an
accomodation; agree on a trustee and you will provide a certain
percentage of each payout to the other party.  Something is better
than nothing - or wait them out.