The Collaborative International Dictionary of English v.0.48:

Primogeniture \Pri`mo*gen"i*ture\ (?; 135), n. [LL., fr. L.
   primus first + genitura a begetting, birth, generation, fr.
   genere, gignere, to beget: cf. F. primog['e]niture, L.
   primogenitus firstborn. See Prime, a., and Genus, Kin.]
   1. The state of being the firstborn of the same parents;
      seniority by birth among children of the same family.
      [1913 Webster]

   2. (Eng. Law) The exclusive right of inheritance which
      belongs to the eldest son. Thus in England the right of
      inheriting the estate of the father belongs to the eldest
      son, and in the royal family the eldest son of the
      sovereign is entitled to the throne by primogeniture. In
      exceptional cases, among the female children, the crown
      descends by right of primogeniture to the eldest daughter
      only and her issue. --Blackstone.
      [1913 Webster]

WordNet (r) 3.0 (2006):

primogeniture
    n 1: right of inheritance belongs exclusively to the eldest son

Moby Thesaurus II by Grady Ward, 1.0:

51 Moby Thesaurus words for "primogeniture":
   aboriginality, age, ancien regime, ancientness, antiquity, atavism,
   bequeathal, bequest, birthright, borough-English,
   cobwebs of antiquity, coheirship, coparcenary, deanship,
   dust of ages, eld, elderliness, eldership, entail, gavelkind,
   great age, heirloom, heirship, hereditament, heritable, heritage,
   heritance, hoary age, hoary eld, incorporeal hereditament,
   inheritance, inveteracy, law of succession, legacy,
   line of succession, mode of succession, old age, old order,
   old style, oldness, patrimony, postremogeniture, primitiveness,
   primordialism, primordiality, reversion, senility, seniority,
   succession, ultimogeniture, venerableness



Bouvier's Law Dictionary, Revised 6th Ed (1856):

PRIMOGENITURE. The state of being first born the eldest.
     2. Formerly primogeniture gave a title in cases of descent to the
oldest son in preference to the other children; this unjust distinction has
been generally abolished in the United States.