We report the observation of the B meson decay B+/- -->J/psietaK+/- and evidence for the decay B0-->J/psietaK0S, using 90 x 10(6) BB; events collected at the Upsilon(4S) resonance with the BABAR detector at the SLAC PEP-II e+e- asymmetric-energy storage ring. We obtain branching fractions of B(B+/- -->J/psietaK+/-) = [10.8 +/- 2.3(stat) +/- 2.4(syst)] x 10(-5) and B(B0-->J/psietaK0S) = [8.4 +/- 2.6(stat) +/- 2.7(syst)] x 10(-5). We search for the new narrow mass state, the X(3872), recently reported by the Belle Collaboration, in the decay B+/- -->X(3872)K+/-,X(3872)-->J/psieta and determine an upper limit of B[B +/- -->X(3872)K+/- -->J/psietaK+/-] < 7.7 x 10(-6) at 90% confidence level.