Exegesis:

egeneto de kai philoneikia en autois ‘an argument also arose among them.’ de kai marks the transition to a new event in the same situation.

philoneikia in a concrete sense, ‘dispute,’ ‘argument,’ ‘quarrel.’

to tis autōn dokei einai meizōn ‘as to who of them was accounted the greatest.’ The interrogative clause is substantivized by prefixing the article to in order to connect it with philoneikia. dokei is used here in the specific sense of being recognized, having the reputation. meizōn means here, ‘greater than the rest,’ cf. Plummer.

Translation:

A dispute … arose among them, or, ‘a quarrel/dissension broke out among them’ (cf. Nieuwe Vertaling; also Tae,’ using a reciprocal form of ‘to be-at-fault’); and cf. on “an argument arose” in 9.46.

Which … was to be regarded as the greatest, or to avoid a passive construction, ‘which … people would consider the greatest,’ ‘who … would have the highest position’; cf. also, “who … should rank highest” (New English Bible). For the greatest see again on 9.46.

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.