What is the origin of the mimetic term じろじろ? As in じろじろ見ないでください。 How does jirojiro mimic or suggest someone staring?
This answer mentions how anime characters will literally say this, but does not explain the ideographic connection to staring.
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The word じろじろ is part of a cluster of related adverbs.
All of these have to do with "moving the eyes and staring pointedly".
I had first thought that maybe the じろ was a shift from しろ and had to do with the whites of the eyes somehow, and the voicing indicated a softer or rounder kind of motion. However, the root is clearly ぢろ, not じろ. I'm not sure how this sound symbolism plays out; the initial ぢ is certainly unusual. It's possible that the initial ぢ arose as a kind of emphasis of じ, in which case this might still be related to the whites of the eyes, and / or to adverb じっと ("fixedly").
The ろ seems to be related to, or at least suggestive of, the ろ in adverbs like ぎょろぎょろ ("rollingly", as when rolling one's eyes) or ころころ ("rollingly", as when something rolls around).