In my opinion the lawyer is mistaken, there is no prohibition on a cousin acting as administrator.

You can get a free consultation from some of the NY probate lawyers listed by location at
http://lawyers.findlaw.com/lawyer/practicestate/Probate-&-Estate-Administration/New-York

http://codes.lp.findlaw.com/nycode/EPT/4/1/4-1.1
has the code section on intestate inheritance in NY.


The property of a decedent not disposed of by will shall be
distributed as provided in this section. In computing said distribution,
debts, administration expenses and reasonable funeral expenses shall be
deducted but all estate taxes shall be disregarded, except that nothing
contained herein relieves a distributee from contributing to all such
taxes the amounts apportioned against him or her under 2-1.8.
Distribution shall then be as follows:
(a) If a decedent is survived by:
///
(6) One or more grandparents or the issue of grandparents (as
hereinafter defined), and no spouse, issue, parent or issue of parents,
one-half to the surviving paternal grandparent or grandparents, or if
neither of them survives the decedent, to their issue, by
representation, and the other one-half to the surviving maternal
grandparent or grandparents, or if neither of them survives the
decedent, to their issue, by representation; provided that if the
decedent was not survived by a grandparent or grandparents on one side
or by the issue of such grandparents, the whole to the surviving
grandparent or grandparents on the other side, or if neither of them
survives the decedent, to their issue, by representation, in the same
manner as the one-half. For the purposes of this subparagraph, issue of
grandparents shall not include issue more remote than grandchildren of
such grandparents.

I hope this information is helpful.