In addition to the biblical use of greater (such as Luke 22; John 13:16; 15;20 ), most extra biblical sorces seem to agree that meizon "greater" has the primary meaning of position.
http://www.forananswer.org/John/Jn14_28.htm
`oti`o pathr meizwn mou estin
hOTI hO PATÊR MEIZÔN MOU ESTIN
for the Father is greater than I.
MEIZÔN
- Greater...of rank and dignity (BADG)
- Pertaining to having a higher status in comparison to something else - 'better, greater, superior to.' (Louw & Nida)
- Greater...in the sense of position not essence (Rogers & Rogers).
- Not a distinction in nature or essence (cf. 10:30), but in rank in the Trinity. No Arianism or Unitarianism here. The very explanation here is proof of the deity of the Son [Dods] (RWP).
Grimm-Thayer, alone among modern lexicons, defines MEIZÔN in such a way as to suggest ontology may be in view: "is used of those who surpass others ... in nature and power, as God: Jn. 10:29, 14:28; Heb. 6:13; 1 Jn. 4:4; add, Jn. 4:12; 8:53" (emphasis added). The question arises whether "nature" in this definition is meant to signify nature of being, as it is used in Trinitarian formulas, or whether it may have a lesser sense - in which case, Grimm-Thayer could then be taken to be in agreement with the other lexicons.
We would first note that in none of the verses cited by Grimm-Thayer does MEIZÔN require the meaning "greater in nature;" on each occasion, the meaning "greater in power, authority, or character" yields a perfectly acceptable interpretation. We may also recall that Thayer translated the Grimm Greek-Latin lexicon into English. The word Thayer translated as "nature" is the Latin natura.
Definition of meizon in Grimm's Lexicon Graeco-Latinum in Libros Novi Testamenti'
citing John 14:28: "meizones dic. qui alios superant vel natura et poteste, ut deus"
It is unlikely that Grimm intended natura to mean "greater in ontological nature." Natura is defined by Lewis & Short as "natural constitution, property, disposition, inclination, temper, or character." Had Grimm intended an ontological meaning, he would probably have used somewhat stronger language, as he does elsewhere when speaking of the Divine Nature (1). To be absolutely clear, he could have used essentia, the Latin term for "nature of being, essence" used commonly in the historic Trinitarian creeds when speaking of the Divine Nature shared by Father and Son (2).
Moulton and Milligan list dozens of extra-Biblical Koine texts, all of which support the "greater in rank or position" definition of MEIZÔN. For example, "MEIZÔN is used in the sense of 'senior' in ostracon receipts...(A.D. 128 [and] A.D. 147).... The word is applied to one in authority, an official...(A.D. 270-275).... 'Occupying a position of highest ... rank and honour' - of a Roman Senator...(c. A.D. 150)." Not one example of ontological greatness is provided.
Thus, the lexical evidence is quite substantial that the meaning "greater in rank or position" is the correct meaning of MEIZÔN.