M. Hilgert, Akkadisch in der Ur III-Zeit, IMGULA 5 (Münster 2002) 76, understands this name as Dīn-Ilī “Judge, oh my god!” or Dīnni-Ilī “Judge me, oh my god!” In note 96, he refers to the Old Akkadian name DI.KU5-i3-li as alternative interpretation. Following W. Sommerfeld, he understands the latter spelling as Dīn-ilē “Judgment of the gods.”
The alternation between DI and di-ku5 in the spelling of the name DI-NI-NI is also found in the text of PDT 2, 1353 and the legend of the seal that is impressed on this tablet. I cannot resolve the inherent difficulties: on the one hand, the equation di-ku5 = dīnu is suspect because di = dīnu, and di-ku5
-ra2, but not di-ku5 could qualify as synonym of di. On the other hand, DI does not seem to be scribal abbreviation for di-ku5= dayyānum “judge,” because TI.NI.NI is found in Ur III texts as a variant spelling of DI-NI-NI.