Considerations revolving around the Little Office are in my mind at the minute; though most of what I've written I've previously thought on years back (I typed up the Dominican Rite Little Office a long time ago, for instance).
Well, what more to say?
It is evident that the main Hours - Matins, Lauds and Vespers - are taken from the Common of Our Lady and from Marian feasts: so the psalmody appointed for Matins and Vespers is that of feasts of the Blessed Virgin.
As for the remaining Hours, for the most part the psalms provided are chosen from the Gradual Psalms (Pss 119-133), except, in the Roman Rite Little Office, for Prime. (In the Dominican Little Office, all fifteen of the Gradual Psalms are used, three each at Prime, Terce, Sext, None and Compline - with the result that the psalms used at Dominican Terce de Beata are the equivalent of those used at Roman Sext de Beata, and so forth.) Why is Prime differently supplied? I conjecture that the Roman choice of Pss 53, 84 and 116 depends upon the truth that, before St Pius X reformed the cursus of the Psalter in 1912, each and every day Prime in the Breviary began with Psalm 53, and so it was felt to be the opening psalm par excellence of Prime, and so needs must be used at Prime of the Blessed Virgin also. I also guess that Psalm 84 is used because its verses well accommodate a Marian and Christological interpretation (hence its special appropriateness in Advent); and Psalm 116 is just a very short psalm of praise added on!
Next, consider the antiphons: in the usual form of the Little Office employed through the year, the psalm antiphons at Matins are from the Common of the Blessed Virgin (and these are also used during Advent and Christmastide, with only one change made during Advent, just as the Breviary prescribes), while at Lauds and through the Hours they are those of the Assumption of Our Lady (Assumpta est... Maria Virgo... In odorem... Benedicta filia... Pulchra es...), while the psalm antiphons for Vespers are simply those from the Common of the Blessed Virgin (Dum esset... Læva ejus... Nigra sum... Jam hiems... Speciosa facta...). Indeed, while considering Vespers, the little chapter (Ecclus xxiv, 14), hymn (Ave maris stella) and collect (Concede nos) are all likewise the same as those appointed for Vespers of feasts of Our Lady; and so too at Lauds, the hymn O gloriosa Virginum is that from the Common also. The hymn at Matins(Quem terra) is from the Common, and the three responsories given for Matins (though the last is rarely used except during Septuagesimatide and Lent, as the Te Deum trumps it) are Sancta et... Beata es... and Felix namque... - and these are Responsories i, iii, and vii (changing festivitatem to commemorationem, and adding a Gloria Patri) from Matins of the Common of feasts of Our Lady. Interestingly, the Invitatory antiphon is from the feast of the Annunciation.
In the Advent form of the Little Office, the psalm antiphons for Lauds, Vespers and through the Hours are those used for the Annunciation on the 25th of March: Missus est... Ave Maria.. Ne timeas... Dabit ei... Ecce ancilla... - all of which are taken from the Gospel of the Annunciation, which the Little Office adopts to read as the Lessons of Matins for the Advent season; and of course the Collect employed at every Hour is Deus qui de beatæ, again from the feast of the Annunciation. Similarly, the three responsories of Matins (Missus est... Ave Maria... Suscipe verbum...) are taken from the first Nocturn of Matins of the Annunciation (Responsories i, ii, and iii). The capitula read at the Advent Little Office day Hours are from Isaias xi, 1-2 (at Lauds, Terce and Vespers - but at None in the Daily Office at the Annunciation); Isaias vii, 14-15 (at Prime and Compline - it is used at Lauds, etc. of the Annunciation in the Breviary); St Luke i, 32 (at Sext just as at the same Hour on the feast of the Annunciation); Ecclus xxiv, 19-20 (at None: this is the only reading that doesn't change in the Little Office, and that doesn't come from the Annunciation feast). It must be noted, by the way, that many of these items proper to the Annunciation also occur or occurred during Advent in the Breviary, and may be originally Advent texts. (This is probably the case for the antiphon Spiritus Sanctus used for all the Gospel canticles for the Little Office in Advent - it is drawn from the Annunciation Gospel, and is somewhat similar to an antiphon used at the Annunciation, but otherwise I haven't tracked down its source.)
During the time from Christmas until the Purification, the antiphons for Lauds, through the Hours and Vespers are those of the Octave Day of Christmas, now quite reasonably styled in the Ordinary Form the Solemnity of Mary, Mater Dei: for the (obviously Byzantine) psalm antiphons appointed are O admirabile... Quando natus... Rubum quem... Germinavit radix... Ecce Maria..., and the Benedictus and Magnificat antiphons Mirabile mysterium... and Magnum hereditatis... (the latter repeated at Compline for the Nunc Dimittis antiphon) are likewise taken from that great feast.
Returning at last to the ordinary form of the Little Office outside of Advent and Christmastide, the texts of Scripture used for lessons and little chapters are nearly all from Ecclesiasticus xxiv, which in the literal sense is the self-praise of the Law of God personified, and in the mystical sense is the self-praise of the Blessed Virgin and of Christ. These are the assigned texts: at Lauds, Cant. vi, 8; at Prime, Cant. vi, 9 (used, slightly modified, for the Benedictus antiphon on the feast of the Assumption); at Terce, Ecclus xxiv, 15 and at Sext, Ecclus xxiv, 16 (the Breviary appoints both passages, joined together, for use at Sext on feasts of the Virgin); at None, Ecclus xxiv, 19-20 (just as for None of Marian feasts according to the great Office); at Vespers, Ecclus xxiv, 14 (used in the Breviary for Marian feasts at Lauds, etc.); and at Compline, Ecclus xxiv, 24. Also, during the course of the year, there are proper Collects at each Hour (and with these may be numbered the Absolution and Blessings at Matins), which are drawn from the large stock of Marian orations in the Church's liturgy (such as those appended to each of the Marian anthems sung after Compline).
All that really remains is to enquire as to the sources of the antiphons for the Benedictus, Magnificat and Nunc Dimittis through the year. Throughout Eastertide, the Regina cæli is employed, that familiar anthem. At Compline, otherwise the ancient Sub tuum (first recorded in Greek in the third century or earlier) is used. At Vespers, a short and rather mediæval anthem is appointed: Beata Mater; and at Lauds, a longer text, said to go back to Augustine (though I suspect a mediæval writer adopting his mantle), the Beata Dei Genitrix.
One last item: the hymn Memento rerum Conditor, used at all five of the Little Hours, has a curious source - for it's a pastiche. The title stanza comes from the Christmas hymn at Vespers by Prudentius (I think), but there is the third stanza! and the middle stanza is a standard mediæval plea to Our Lady, with the third being the standard doxology employed on all Marian feasts. Apparently in mediæval times, a stanza from the Veni creator Spiritus was also inserted, in honour of the Holy Ghost, Who also had a commemoration (antiphon, verse and collect) at each and every Hour.
Enough for now.
P.S. It really ought be noted that the Little Office according to the Use of Rome as we have it to-day is apparently somewhat reworked compared to the many variant forms of the Little Office in mediæval times. I recall reading that at the time St Pius V brought out the Roman Breviary (containing the Little Office), some changes were made in it in accord with the liturgical spirit of the times. The Dominican Rite Little Office, as given in an earlier posting, is much more mediæval in its antiphons, for instance, and especially in its provision of three prayers to Our Lady as Lessons for Matins - these three prayers were exceedingly commonly used in mediæval versions of the Little Office, and I think they bear repeating here; note how they rhyme:
Sancta Maria, Virgo virginum,
mater et filia Regis regum omnium,
tuum nobis impende solatium:
ut cælestis regni per te mereamur habere præmium,
et cum electis Dei regnare in perpetuum.
Sancta Maria, piarum piissima,
intercede pro nobis, sanctarum sanctissima:
per te, Virgo, nostra sumat precamina
qui pro nobis ex te natus regnat super æthera:
ut sua caritate nostra deleantur peccamina.
Sancta Dei Genetrix, quæ digne meruisti concipere
quem totus orbis nequit comprehendere,
tuo pio interventu culpa nostras ablue:
ut redempti perennis sedem gloriæ
per te valeamus scandere,
ubi regnas cum eodem Filio tuo sine tempore.