ἀναίδεια, ας, ἡ (...) persistence, impudence, lit. shamelessness Lk 11:8 ...
Arndt, W., Gingrich, F. W., Danker, F. W., & Bauer, W. (1979). In A Greek-English lexicon of the New Testament and other early Christian literature : a translation and adaption of the fourth revised and augmented edition of Walter Bauer’s Griechisch-deutsches Worterbuch zu den Schrift en des Neuen Testaments und der ubrigen urchristlichen Literatur (p. 54). University of Chicago Press.
There seems to be a sense of all three of these meanings involved here. Whenever Jesus taught persistence in prayer, he did so with including a sense of urgency and utter sincerity.
And he told them a parable to the effect that they ought always to pray and not lose heart. 2 He said, “In a certain city there was a judge who neither feared God nor respected man. 3 And there was a widow in that city who kept coming to him and saying, ‘Give me justice against my adversary.’ 4 For a while he refused, but afterward he said to himself, ‘Though I neither fear God nor respect man, 5 yet because this widow keeps bothering me, I will give her justice, so that she will not beat me down by her continual coming.’ ” 6 And the Lord said, “Hear what the unrighteous judge says. 7 And will not God give justice to his elect, who cry to him day and night? Will he delay long over them? 8 I tell you, he will give justice to them speedily. Nevertheless, when the Son of Man comes, will he find faith on earth?”
(Luke 18:1–8, ESV)
He condemned the persistency of empty repetition:
“And when you pray, do not heap up empty phrases as the Gentiles do, for they think that they will be heard for their many words.
(Matt 6:7, ESV)
Jesus also condemned the self-righteous prayer, the prayer that ignores asking for what one really needs; as in Matt. 6, the prayer to impress other people than in secret to God.
He also told this parable to some who trusted in themselves that they were righteous, and treated others with contempt: 10 “Two men went up into the temple to pray, one a Pharisee and the other a tax collector. 11 The Pharisee, standing by himself, prayed thus: ‘God, I thank you that I am not like other men, extortioners, unjust, adulterers, or even like this tax collector. 12 I fast twice a week; I give tithes of all that I get.’ 13 But the tax collector, standing far off, would not even lift up his eyes to heaven, but beat his breast, saying, ‘God, be merciful to me, a sinner!’ 14 I tell you, this man went down to his house justified, rather than the other. For everyone who exalts himself will be humbled, but the one who humbles himself will be exalted.”
(Luke 18:9–14, ESV)
Jesus' mother made a request to Jesus which is a good example of διά τὴν ἀναίδειαν, although the phrase is not used here.
When the wine ran out, the mother of Jesus said to him, “They have no wine.” 4 And Jesus said to her, “Woman, what does this have to do with me? My hour has not yet come.” 5 His mother said to the servants, “Do whatever he tells you.”
(John 2:3–5, ESV)
In both cases the embarrassment from the lack of resources was involved, but the request was made to someone who had the resources though not convenient to supply.
Roberston referenced more examples:
Yet because of his importunity (δια γε την ἀναιδιαν αὐτου [dia ge tēn anaidian autou]). From ἀναιδης [anaidēs], shameless, and that from α [a] privative and αἰδως [aidōs], shame, shamelessness, impudence. An old word, but here alone in the N. T. Examples in the papyri. The use of γε [ge] here, one of the intensive particles, is to be noted. It sharpens the contrast to “though” by “yet.” As examples of importunate prayer Vincent notes Abraham in behalf of Sodom (Gen. 18:23–33) and the Syro-Phoenician woman in behalf of her daughter (Matt. 15:22–28). -- Robertson, A. T. (1933). Word Pictures in the New Testament (Lk 11:8). Broadman Press.
All these cases show an urgency on the part of the requestor.
The urgent request of a righteous person is very powerful in its effect.
(James 5:16b, HCSV)
When it comes to importunity in prayer, we can look at positive examples of importunity in the Scriptures. What stands out is boldness, e.q. Esther (Esther 4:10-11) and the Syro-Phoenician woman (Mark 7:25-30). In many languages, when entering a superiors office or interrupting a conversation, one starts with a phrase that essentially means, "Pardon my rudeness." Because of Christ's sacrifice, we can approach God with boldness (Heb. 4:14, ESV).