The noun pupil denotes the dark circular opening in the centre of the iris of the eye, which varies in size to regulate the amount of light reaching the retina.

Via the Middle-French feminine noun pupille, this word is from classical Latin pūpilla. This Latin feminine noun was a diminutive of pūpa, meaning literally a girl, and in transferred use also a doll (cf. the French feminine noun poupée, a doll). The word pūpilla, meaning a little girl, a little doll, came to also denote the pupil of the eye on account of one’s own reflection seen when looking into somebody’s eye (cf. also the apple of one’s eye – la prunelle de ses yeux). The Roman statesman and scholar Pliny the Elder (23-79) evoked this phenomenon in Naturalis Historia (The Natural History – 77), a vast encyclopaedia of the natural and human worlds:

In the midst of the cornea of the eye Nature has formed a window in the pupil, the small dimensions of which do not permit the sight to wander at hazard and with uncertainty, but direct it as straight as though it were through a tube, and at the same time ensure its avoidance of all shocks communicated by foreign bodies. The pupils are surrounded by a black circle in some persons, while it is of a yellowish cast with others, and azure again with others. By this happy combination the light is received by the eye upon the white that lies around the pupil, and its reflection being thus tempered, it fails to impede or confuse the sight by its harshness. So complete a mirror, too, does the eye form, that the pupil, small as it is, is able to reflect the entire image of a man. This is the reason why most birds, when held in the hand of a person, will more particularly peck at his eyes; for seeing their own likeness reflected in the pupils, they are attracted to it by what seem to be the objects of their natural affection.
translation: John Bostock and Henry T. Riley (1855)

The sense development was identical in ancient Greek: κόρη (= kόrē), literally a girl, a maiden, came to also denote the pupil of the eye.

The Latin feminine form pūpilla corresponded to the masculine pūpillus, a variant of pūpulus, literally a little boy, in turn a diminutive of pūpus, a boy, a child. Both pūpilla and pūpulus also designated an orphan, a ward, a minor. The English noun pupil in the sense of a person who is being taught by another was originally a legal term denoting an orphan who is a minor and consequently a ward, which is the still the acceptation of the French noun pupille, used either in the masculine or in the feminine (whereas the homonymic feminine noun pupille denotes the pupil of the eye).

A sense development identical to those of Latin pūpilla and Greek κόρη explains the use of English baby in the phrase to look babies in somebody’s eyes, originally meaning to gaze at one’s own reflection in somebody’s eyes, later also to gaze closely and lovingly into somebody’s eyes.

This phrase is first recorded in Syr P. S. His Astrophel and Stella. Wherein the excellence of sweete Poesie is concluded. To the end of which are added, sundry other rare Sonnets of divers Noble men and Gentlemen (London, 1591), by Philip Sidney (1554-86), English author and courtier:

When thou sawest in Natures cabinet,
Stella, thou straight look’st babies in her eyes.

The metaphor also appeared in Tell-Trothes New-yeares Gift Beeing Robin Good-fellowes newes out of those Countries, where inhabites neither Charity nor honesty. With his owne Inuective against Ielosy (London, 1593):

(1876 reprint)
Loue is a pleasing gout, which will suffer vs no more to be misled by vnrest, then [= than] the tormenting gout wil giue his patientes leaue to rest while the paine is vnceasing. And such a hartie dropsie is he, as he swels his criples affections with so great kindnesse, as they sing no song, but Ah, I loue. He is a nettle that stinges the hart with continuall pleasure; and that babie which lodges in womens and mens eies, on whome none shall fix the fancy kindly, that shall not be stroken with a darte of constauncy; hee is the greeuing woe that breedes continuall ioy, the fond conceipt that fastens faithful thoughts in his place, and that euill that reapes eternall good.

In Room at the Top (London, 1957), by the English novelist John Gerard Braine (1922-86), childhood memories are awakened in Joe, the narrator, when Alice elaborates on the phrase:

I looked into her eyes. I could see my face in her pupils, flushed, with my hair tousled. ‘You’re looking babies,’ she said, almost coyly. ‘If you look long enough, you’ll see a baby’.
I had the same sensation that I had when as a child of ten I’d seen my Aunt Emily with her son at her breast. And it was, too, like the sensation I’d had when I’d intercepted looks and actions of my parents – the secret, bold look before bedtime, the hand on the knee.