Exegesis:

eisēlthen de dialogismos en autois lit. ‘a discussion, or, a thought entered into them,’ hence—in order to fit dialogismos (either meaning, see below)—‘came about,’ ‘happened.’

dialogismos ‘reasoning,’ ‘deliberation,’ either outward and audible, i.e. ‘discussion,’ ‘argument,’ or inward and inaudible, i.e. ‘consideration (in oneself),’ ‘thought,’ ‘doubts.’ Since a contrast with ton dialogismon tēs kardias (‘reasoning of the heart,’ hence ‘thought’) in v. 47 appears to be present, the former rendering is preferable here.

en autois ‘among them.’

to tis an eiē meizōn autōn ‘as to which of them was the greatest’ or ‘as to who might be greater than they,’ indirect question indicating what the argument was about, introduced by the article to, cf. on 1.62. autōn is best understood as a partitive genitive, going either with tis (‘which of them’) or with meizōn (‘the greatest of them’), preferably the former. meizōn is to be interpreted as a superlative.

Translation:

An argument arose among them, or, ‘a deliberation grew/was-born among them’ (Batak Toba), “they fell into an argument” (The Four Gospels – a New Translation), ‘they disagreed-with-each-other’ (Toraja-Sa’dan), ‘they fought in speech’ (Sundanese). The expression is not neutral, as is “said to one another” (see on 4.36), but it implies less animosity than “a dispute arose” (22.24).

As to which of them was … The indirect question may have to become a direct one, ‘which of us may be…?,’ or, ‘which of us is the greatest, do you think?’ Greatest. Some languages use here ‘most in front’ (Javanese, Balinese), ‘the highest’ (Sranan Tongo); see also on 1.15. The superlative may be rendered by ascribing the qualification exclusively to one of the group, e.g. ‘of/among all of them, which one (is) great.’

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.