"On that night the king's sleep stayed away from him" (Esther 6:1 - translation follows Radak).
Why does the passuk use a lashon of nadedah, "stay away," instead of the more simple wording, "On that night the king couldn't sleep"?
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The root of "נדדה" denotes movement and, as you translated, distancing. Both of these contribute to Radak's generalization of the root's meaning in the Sefer Hasharashim (here), "כולם ענין התנועה וההרחקה". Of the two other references he gives that pertain directly to sleep/night activity one is from1 Ya'akov's self-report of the hardship of his life in the Lavan regime. He says
הָיִיתִי בַיּוֹם אֲכָלַנִי חֹרֶב וְקֶרַח בַּלָּיְלָה וַתִּדַּד שְׁנָתִי מֵעֵינָי
Often, scorching heat ravaged me by day and frost by night; sleep וַתִּדַּד from my eyes.
There, it is not a matter of interpretation that the word denotes a tortured and unrestful sleep rather than passively being awake at the wrong time. That is Ya'akov's point.
It may be fair to impute the same mode of sleep disturbance to Achashverosh. Back in the realm of interpretation, this could explain how some of the detailed midrashim on the untold events of this episode were informed. The g'mara in M'gila 15b understood that Achashverosh was greatly troubled over the possibility of his assassination and a missed opportunity for redress in his recent past, which led to his request to review the logs.
נפלה ליה מילתא בדעתיה אמר מאי דקמן דזמינתיה אסתר להמן דלמא עצה קא שקלי עילויה דההוא גברא למקטליה הדר אמר אי הכי לא הוה גברא דרחים לי דהוה מודע לי הדר אמר דלמא איכא איניש דעבד בי טיבותא ולא פרעתיה משום הכי מימנעי אינשי ולא מגלו לי מיד ויאמר להביא את ספר הזכרונות דברי הימים
Rashi is likewise explicit about the fact that he was being tormented by justice-seeking amends-makers:
שהיו מלאכים מבהילים אותו כל הלילה ואמרו לו כפוי טובה שלם טובה למי שעשאה
So it could be that נ.ד.ד was the most descriptive verb for the violent nature of Achashverosh's lack of sleep (which would fit nicely with the oscillation/flapping sense of several appearances of this root if it somehow hints at tossing and turning in one's sleep, but I have no source for that connection).
1 The other one is from Iyov and therefore less likely to be informative on its own: אִם־שָׁכַבְתִּי וְאָמַרְתִּי מָתַי אָקוּם וּמִדַּד־עָרֶב וְשָׂבַעְתִּי נְדֻדִים עֲדֵי־נָשֶׁף