Mark 15:12 reads quite nicely in English translations and it is not too difficult to figure out what the verse is saying in Greek. Describing it grammatically, however, is a bit of a tangle, especially when an accounting of every element must be rendered—as is the case in the Baylor Handbook series. Here’s my first draft entry for the verse along with a translation and a grammatical diagram.
15:12 ὁ δὲ Πιλᾶτος πάλιν ἀποκριθεὶς ἔλεγεν αὐτοῖς, Τί οὖν [θέλετε] ποιήσω [ὃν λέγετε] τὸν βασιλέα τῶν Ἰουδαίων;
ὁ … Πιλᾶτος. Nominative subject of ἔλεγεν.
πάλιν. “Again” because Pilate has already addressed them in this regard: ὁ δὲ Πιλᾶτος ἀπεκρίθη αὐτοῖς λέγων (v. 9).
ἀποκριθεὶς. Aor mid ptc masc nom sg ἀποκρίνομαι (means). Redundant adverbial participle of speaking; see the discussion of this intransitive, “-θη- middle” form at 3:33.
ἔλεγεν. Impf act ind 3rd sg λέγω.
αὐτοῖς. Dative indirect object of ἔλεγεν.
Τί οὖν [θέλετε] ποιήσω [ὃν λέγετε] τὸν βασιλέα τῶν Ἰουδαίων; Clausal complement (direct discourse) of ἔλεγεν.
Τί. Interrogative pronoun, accusative direct object of θέλετε.
θέλετε. Pres act ind 2nd pl θέλω.
ποιήσω [ὃν λέγετε] τὸν βασιλέα τῶν Ἰουδαίων. Clausal complement of θέλετε. One might have expected this clause to have been introduced by ἵνα (though that is a rare construction; it is not used in NT or LXX, but see Hist. Rech. 7:13, τί θέλεις ἵνα ποιήσωμέν σοι;), but it is intelligible as it stands: it indicates the content of Pilates’ query as to what the crowd (really, the chief priests) wanted (θέλετε) him to do. The pattern “interrogative + second person indicative of θέλω + aorist subjunctive” is the usual way to express this (see Matt 27:17, 21; Mark 10:36; L.A.E. 29:2; 4 Bar. 3:9).
ποιήσω. Aor act subj 1st sg ποιέω. Here the sense is “to do to/with” (BDAG, 841.4).
[ὃν λέγετε] τὸν βασιλέα τῶν Ἰουδαίων. Clausal complement of ποιήσω.
ὃν. Accusative direct object of λέγετε in an object-complement double accusative construction. Matthew’s construction is similar, though substituting the name for the pronoun and a synonymous expression for the complement: Τί οὖν ποιήσω Ἰησοῦν τὸν λεγόμενον Χριστόν; (Matt 27:22).
λέγετε. Pres act ind 2nd pl λέγω. In this context the verb means “to call, name” (BDAG, 590.4). In placing these words in the mouth of the crowd (second person), Pilate does not say that Jesus is such a king, but only that they have used that appellation. He may, in doing so, be mocking them. In Mark’s account neither the crowd nor the chief priests have so designated Jesus, but that the religious leaders have accused him of claiming to be a king is implied in Pilate’s question in 15:2. Luke 23:2 confirms that this was, indeed, one of the charges leveled against Jesus.
τὸν βασιλέα τῶν Ἰουδαίων. Accusative complement in an object-complement double accusative construction. τῶν Ἰουδαίων is an objective genitive indicating those over whom the king reigns.
So Pilate responded to them again, “What, then, do you want me to do with the one whom you call ‘The King of the Jews’?”
(No, the Baylor Handbook will not have grammatical diagrams, though it would be helpful to conceptualize sentences such as this!)