Games in which a Mets player was a single short of the cycle:

  Player            Date          Tm   Opp GmReslt PA AB  R  H 2B 3B HR RBI BB
 +-----------------+-------------+---+----+-------+--+--+--+--+--+--+--+---+--+
  Gregg Jefferies   1989-07-22    NYM  ATL W  7-5   4  4  2  3  1  1  1   2  0
  Gregg Jefferies   1988-09-09    NYM @MON W  7-3   5  5  1  3  1  1  1   2  0
  Gregg Jefferies   1988-08-29    NYM  SDP W  6-0   4  4  2  3  1  1  1   2  0
  Darryl Strawberry 1987-08-16    NYM @CHC W 23-10  6  5  5  4  2  1  1   5  1
  Ron Swoboda       1967-06-06(2) NYM @PIT W  3-2   4  4  2  3  1  1  1   2  0
  Joe Christopher   1964-08-18    NYM  PIT W  7-3   5  5  2  4  1  2  1   3  0

Interesting that Gregg Jefferies charted three of the six games, all within the span of eleven months covering two seasons. The Mets won all six of these games. On two occasions, a Met collected the requisite four hits for a cycle, but substituted an extra-base hit for the single. In fact, in the 8/18/1964 game, Joe Christopher collected two triples to go along with the double and homerun, but nary a single.

245 times a Met has come up a triple short of a cycle.
69 times a Met has come up a homerun short of a cycle.
26 times a Met has come up a double short of a cycle.

Stats courtesy Baseball-Reference.com's Play Index.