Redundantia originally conveyed the image of waves -- after one wave, another wave comes, and this re-undulation is what redundantia originally meant. You can see how this would connect to our word 'redundancy'. Historically, however, the word meant a superabundance in a broader sense, and not necessarily a mere redundancy; it suggests being overwhelmed (you manage to get through one wave and another hits you) or having more than you know what to do with. Usually you will find it and its cognates translated as 'overflow'. I would suggest that redundantia in a slightly more technical -- but still connected -- sense is an important idea in the thought of Aquinas that has almost universally been overlooked.
Aquinas uses the term, or cognates, in a wide number of circumstances, but all of them have important links to each other. I'll just briefly note a few cases.
Divine order is such that excellence overflows from higher to lower, like the clarity of sun into the air: ST 2-2.83.11. Compare ST 2-2.175.2 ad 1, which uses the related word abundantia in a quotation from Dionysius on the overflow of divine goodness, thus linking it to the idea of participation.
II. Human Cognition
Desire in the superior part of the soul can be so vehement that it overflows lower desire, so that the latter, in its own way, tends toward the spiritual good of the higher desire, so that the body serves the spiritual: ST 2-1.30.1 ad 1.
In the human soul there is an overflow from higher to lower, so that the delight of contemplation can overflow so as to mitigate sensible pain or sorrow: ST 2-1.38.4 ad 3.
Human cogitative and memorative powers owe their excellence (compared to the corresponding capabilities in other animals) simply to their association with reason, which in a way overflows them: ST 1.78.4 ad 5.
Vocal prayer is an overflow from the soul to the body through a vehemence of affection: ST 2-2.83.12
The qualities of prudence overflow into all other virtues, and likewise with the other cardinal virtues: 2-1.61.4 ad 1.
Because lower powers follow the motion of higher powers, the states of intellectual desire following from virtues overflow into sensible desire to cause passions; so that if joy in the will is increased through justice, it will cause the passion of joy in the sensible appetite: ST 2-1.59.5.
IV. Christ in the Transfiguration
That Christ's glory did not overflow His body from conception was due to a special dispensation, but he retained the power of pouring out the glory of His soul into His body; in Christ's transfiguration, the glory of His Godhead and His soul overflowed His body not as an innate quality but as a transient passion, like the clarity of the sun in the air, and thus miraculously: ST 3.45.2.
Since Christ was still on earth, in His Passion there was no overflow of glory from the higher part of his soul to the lower part, nor from his soul to his body, but the higher part of his soul still enjoyed contemplation of God: ST 3.46.8.
V. Christ as Head of the Church
Grace was bestowed on Christ not merely as an individual but as Head of the Church, so that it might overflow into His members: ST 3.48.1. Christ as man is mediator between God and men, so it is appropriate for him to have a grace that overflows to others: ST 3.7.1. Christ was predestined to be the Son of God in power of sanctification, so it was appropriate for Him to have a fullness of grace overflowing to all, while the Virgin Mary is full of grace as being close to Him and dispensing grace to us by receiving Him: ST 3.27.5 ad 1. The Virgin Mary was full of grace in that it overflowed from her soul to her body, from which was conceived the Son of God; and it is such as to extend to all: Exp. Sal. Ang. art. 1.
To know the secrets of the heart belongs properly to God alone, but Christ Ascended may also know and judge human hearts as man through an overflow from His Godhead: ST 3.59.2 ad 3.
VI. Glory to Come
The soul desires to enjoy God in such a way that its enjoyment may overflow into the body, as far as this is possible: ST 2-1.4.5 ad 4.
Some people attribute Christ's Ascension to the glorified soul itself, whose overflow glorifies the body, as Augustine suggests (Ep. ad Dioscor. cxviii); but as the body is made glorious by participating the soul, so the soul is made glorious by participating God, and thus divine power is the first cause of the Ascension: ST 3.57.3.
Paul's vision of God in the third heaven did not beatify him so as to overflow his body, but only incidentally: ST 2-2.175.3 ad 2.
By divine ordinance, glory overflows from the soul to the body according to merit, so that as we merit by acts of the soul in the body, we are rewarded by the glory of the soul overflowing the body: ST 3.19.3 ad 3. Our bodies are unable to enjoy God by directly knowing and loving Him, but it is through bodily acts that we attain to knowledge of God; so that the enjoyment of the soul overflows into the body with a 'flush of health and incorruption', as Augustine says: ST 2-2.25.5 ad 2. Charity extends to that from which happiness flows, namely, God, and to that which directly perceives happiness, namely, men and angels, and to that to which happiness comes by a kind of overflow, namely, the body: ST 2-2.25.12. Friendship based on full participation in happiness, as with love of neighbor, is a greater reason for love than that based on happiness by overflow, as with love of our own body: ST 2-2.26.5.
The clarity of the soul overflows the glorified body by way of a permanent quality: ST 3.45.2.
There are many other passages that could be added. Interestingly, Aquinas does not seem to apply the concept in the one situation in which modern Catholics are likely to come across it being applied: the sacrament of the Anointing of the Sick. Unction is a sacramental preparation of the soul. Historically it has been associated with all sorts of healings and recoveries, but these have not generally been considered miracles in the proper sense; rather, they are understood as being the result of overflow: as we take courage and consolation from the sacrament, this allows our body to rally, at least temporarily, by taking part in the mental surge and comfort. It is, as we would say, 'psychosomatic'. That it should be understood in this way is unsurprising, I think. First, because all the sacraments and sacramentals involving oil indicate, by the very use of oil, that some kind of overflow is expected or asked for. And second, and most importantly, all the major sacraments are associated with some major aspect of Church doctrine (Baptism with the Trinity, Eucharist with the Incarnation, Matrimony with the Church, and so forth); the aspect of Church doctrine with which Unction is associated is the Resurrection of the Dead and Life in the World to Come. So the oil is a sign of the overflow of grace into the soul through Christ our Head, which may and sometimes does overflow the body, because strength and consolation of mind are already capable of overflowing the body; and this all is itself a sign of the overflow of glory from God to the soul to the resurrected body in the life to come. Aquinas, as I said, doesn't see this connection (although we have to keep in mind that we never get his final account, since it would have been in the part of the Summa that was never finished). What he says is consistent with it, but his account depends on an analogy with the sacrament of Baptism (Baptism is to the internal as Unction is to the external), and not on the redundantia which later Catholics came to recognize in it, and which was occasionally suggested in various versions of the rite. (It would be interesting to look at the history of this development.)