Here is some academic references:
Bearer of a fair offspring, name by which Demeter was invoked in the Thesmophoria, Ar.Th.299, Alciphr.2.4, cf.IG14.205 (Acrae); or her nurse, Ar. ap. Phot.; epith. of the Moon, Hymn.Mag.5.31; of the Earth, Apollod. ap. Phot.:—neut. pl., Καλλιγένεια θύειν offer sacrifice to Demeter K., Alciphr.3.39 (nisi leg. τῇ Κ.).
Source: Henry George Liddell, Robert Scott, A Greek-English Lexicon (καλλι-γένεια)
Although Καλλιγενεια has been used to refer to a nursemaid serving Demeter, that usage smacks of poetic invention. I find Καλλιγενεια as a name of Demeter (or an analog), used to invoke her during the autumn festival, much more credible as the original usage.
The fair offspring referred to would be Persephone, daughter of Demeter, who goes to the underworld during winter, and whose return brings the spring. Persephone is most fair because without her return, everything on earth dies.
From the Aristophanes:
Silence! Silence! Pray to the Thesmophorae, Demeter and Cora; pray to Plutus, Calligenia, Curotrophus, the Earth, Hermes and the Graces, that all may happen for the best at this gathering, both for the greatest advantage of Athens and for our own personal happiness! May the award be given her who, by both deeds and words, has most deserved it from the Athenian people and from the women! Address these prayers to heaven and demand happiness for yourselves. Io Paean! Io Paean! Let us rejoice!
SOURCE: Aristophanes, Thesmophoriazusae
Here Calligenia definitely seems to be a separate entity, but the source is a comic play, and playwrights are known for "innovations" in myths. Note that the Thesmophorae are dual, both Demeter and her daughter.
"Cora" refers to Persephone's other name, Kore (κόρη) which means simply "girl". Prior to her sacred marriage to Hades each year that hearkens winter, Kore resets to her maiden aspect with her rebirth in the spring. Persephone is both maiden and crone, depending on the time of the year.
The second available fragment mentioning Kallegenia as a nurse maid of Demeter, as opposed to Demeter herself, comes from the 5th Century AD, ~800 years after Aristophanes:
There also she left Kalligeneia, her own fond nurse [to care for Persephone], with her baskets, and all that cleverhand Pallas gives to make womankind sweat over their wool-spinning."
Dionysiaca VI.140 (English) | (Greek)
The lexical entry for Kallegeneia lists a few other sources, such as Alciphron, somewhere between Aristophanes and Nonnus, who seems to regards Kalligeneia as an epithet for Demeter.
The main reason I regard Aristophanes' take to be unsatisfactory is the construction of the name Kalligeneia, and the meaning of "bearer" in the name:
This is distinct from:
- φέρω (fero), which is the general word for carry.
It is in the sense of fero that a nursemaid carries an infant, but it is only the mother who carries in the sense of gen-.
γενεά (genea): offspring
γενεῆθεν (genethen): from birth, by descent
γενέτειρα (genetayra): mother
Iphigenia (Ἰφιγένεια) is an example of a name with the gen- root. Although there are multiple proposed meanings (either "born of strength", "mothering a strong race" or "she who causes the birth of strong offspring",) all meanings relate to birth or parentage.