The BDAG lexicon offers a lengthy discussion of κρίνω, with usages ranging from "to make a selection" (with glosses select, prefer) and on to "to pass judgment upon," whether in cognitive senses (with glosses such as decide, think, consider, etc.) or in judicial senses (with glosses such as judge, decide, condemn, etc.). The very last usage, "to ensure justice for someone" (gloss see to it that justice is done), lists only 1 Clem 8.4 and Isa 1.17 [LXX] as examples. Louw and Nida don't offer an equivalent for this last usage; they give the glosses decide, prefer, evaluate, hold a view, make legal decision, condemn, and rule. But they list Luke 22.30 under this last domain, rendering the passage, "and you will be seated upon thrones, ruling over the twelve tribes of Israel" (§37.49, my emphasis). In their view, then, κρίνω here involves far more than simply "judging." BDAG discusses this verse under the usage "to engage in a judicial process" (with the glosses judge, decide, hale before a court, condemn, and hand over for judicial punishment [see §5.b.β]), saying "occupied by those who have been divinely commissioned to judge: the 12 apostles judge the 12 tribes." They do add the caveat, "But here κ[ρίνω] could have the broader sense rule" (s.v.), but their preference is clear.
I find myself siding, with some fear and trepidation, against BDAG and with Louw and Nida. I am largely convinced by Richard Horsley's argument, in Jesus and the Spiral of Violence (Minneapolis: Fortress Press, 1987, 199–207), that this passage should be translated, "You yourselves will also sit on twelve thrones saving (or effecting justice for) the twelve tries of Israel." Horsley doesn't discussion Matt 12.18, 20 in this regard, but he could have. In that passage the evangelist cites a lengthy passage from Isaiah 42 (in a text-form closer to our MT readings than to the LXX), which twice uses the cognate noun κρίσις [krisis] clearly in the sense of "justice":
καὶ κρίσιν τοῖς ἔθνεσιν ἀπαγγελεῖ. . . . ἕως ἂν ἐκβάλῃ εἰς νῖκος τὴν κρίσιν.
kai krisin tois ethnesin apangelei. . . . heōs an ekbalē eis nikos tēn krisin.
And he will proclaim justice to the nations. . . . until he shall bring justice into victory. (Matt 12.18, 20)
I don't think Matthew 12 makes the case for Matt 19.28 (and certainly it doesn't for Luke 22.28–30), but I think it at least gives us a solid basis for understanding the κριν- root in terms of justice or restoration.
Of course, the difference between judging and effecting justice for is fairly dramatic. And of course we want to translate the passage in a way that communicates accurately the text's meaning. But here we can't deny that the act of translating the Greek is part of determining this verse's meaning (rather than simply communicating its meaning). This, of course, is always the case, but in this instance at least the stakes seem to be that much higher.
So what do you think? How should we render κρίνω [krinō] in Matt 19.28||Luke 22.28–30? What does Jesus tell his disciples they will do sitting upon the twelve thrones of Israel?
[Update: In my lecture notes I've rendered Matt 19.28, "Then Jesus said to them, 'Truly I tell you that, in the age of renewal, when the Son of Man sits upon his throne of glory, you who have followed me will also sit upon twelve thrones and mete out justice for the twelve tribes of Israel.'"]