In 1858 St. Bernadette asked for the name of the beautiful Lady who was appearing to her. The Beautiful Lady answered her "I am the Immaculate Conception." Why didn't she simply tell her she was the Mother of God? Or why didn't she say "I am immaculately conceived."
Our Lady of Lourdes appears to St. Bernadette and tells her "I am the Immaculate Conception".
Joan Cruz's See How She Loves Us: 50 Approved Apparitions of Our Lady, § "Our Lady of Lourdes":
Members of the family and countless visitors had implored Bernadette several times to ask the identity of the Lady. Bernadette had done so, but the Lady had not responded. During the sixteenth apparition on March 25, 1858, the Feast of the Annunciation, Bernadette asked once more. This time the Lady responded in the Bigourdane dialect, “Que soy era Immaculada Conceptiou,” which means, “I am the Immaculate Conception.”
In the year 1854, four years before the start of the apparitions at Lourdes, Pope Pius IX had declared [in Ineffabilis Deus] the doctrine of the Immaculate Conception of the Virgin Mary to be an “article of faith” to be believed by all Catholics. The doctrine was neither understood nor discussed among persons of Bernadette’s station in life,* so it was reasonable that Bernadette did not understand the words spoken by the Lady.
*Her family was extremely poor, living in a barn with a window facing a dunghill.
St. Bernadette—who was 14 years old, unlettered, and failed her catechism exam—couldn't even pronounce "Immaculada Conceptiou" when she told others this was who the Lady said she was. However, priests certainly knew that the Immaculate Conception is the Mother of God. There is no way St. Bernadette could have fabricated this, which proves the miraculous nature of the Our Lady of Lourdes apparitions.
Why she said "I am the Immaculate Conception" and not "I am immaculately conceived"
Fr. Karl Stehlin's Who Are You, O Immaculata? p. 114 (PDF p. 132) explains:
At Lourdes the Immaculate Virgin answered St. Bernadette: ‘I am the Immaculate Conception!’ With these words she clearly stated that she is not only immaculately conceived, but is the Immaculate Conception. In the same way, a thing that is white is something other than whiteness itself, and a perfect thing is something other than perfection itself. When God spoke about Himself, He said to Moses, ‘I am Who am,’ that is, it belongs to My essence that I always have my being from Myself, without beginning. In contrast, the Immaculate Virgin has her beginning in God, is a creation, is a conception. Nevertheless she is the Immaculate Conception.
We can say likewise that she is the greatest, most excellent, purest Temple of the Holy Ghost. Mary herself corroborates this truth when she defines herself at Lourdes: “I am the Immaculate Conception” and thus assigns to herself the title that in the strict sense is an attribute of God (“I am...”) and is applied in particular to the Holy Ghost, who within the Trinity is the eternally perfect, “immaculate” conception of the Father and the Son.
If among creatures a bride takes the name of her husband by the fact that she belongs to him, unites herself with him, makes herself like unto him and together with him becomes the source of new life, how much more should the name of the Holy Spirit, "Immaculate Conception", be the name of her in whom He lives with a love which is fruitful in the entire supernatural economy?4
4. Final article of February 17, 1941, KR 212-213
(taken from this answer)
Why did Our Lady of Lourdes say “I am the Immaculate Conception” and not “I am immacutalely conceived” or “I am the Mother of God”?
The answer is very simple, it would be wrong if Blessed Virgin Mary would answer the question of St.Bernadette asking "Who are you?" with a description "I am immaculately conceived". There's no way St.Bernadette can identify the Blessed Virgin Mary with that answer.
Next, if the Blessed Virgin Mary answered St.Bernadette with "I am the Mother of God" it would negate the purpose of the apparition which is very much related to previous apparition of Mary to St.Catherine of Laboure with a prayer invocation;
"O Mary conceived without sin pray for us who have recourse to thee."
The Blessed Virgin Mary answered St.Bernadette question with the words "I am the Immaculate Conception" is the truthful answer because Mary was not the one speaking but a Divine Person. Mary was possessed by the Holy Spirit and St.Maximillian Kolbe has a good explanation on this mystery.
Please see my full answer on the subject of "quasi-incarnate". Is "quasi-incarnation" been attributed to mystical union of the Holy Spirit and Mary?
In closing the Ineffabilis Deus defined that Mary was preserved from all stains of original sins. But Mary's revelation saying "I am the Immaculate Conception" speaks of a person revealing Herself thru Mary's humanity was in no way present to the contents of the Dogma of Immaculate Conception. In a way this is a profound revelation and St.Maximillian Kolbe gave us the a good reflection how to grasp the meaning of the revelation "I am the Immaculate Conception.
The nature of the union [of the Holy Ghost and Mary] consists in the union of wills. Mary identifies so thoroughly with the will of God that one can speak about a quasi-incarnation of the Holy Ghost in Mary.
The Third Person of the Most Blessed Trinity was not made flesh. Yet our human word "spouse" cannot express the reality of the relation between the Immaculata and the Holy Ghost. We can therefore say that the Immaculata is in a certain sense an "incarnation of the Holy Ghost". The Holy Ghost, whom we love, is in her, and through her we love the Son. The Holy Ghost is very little appreciated.
- Maximilian Kolbe always insisted that of course there can be no question here of a real incarnation of the Holy Ghost, which would be heretical. Instead he is searching for words and concepts that portray more profoundly the intimate relation between Mary and the Holy Ghost. Therefore the qualifier “quasi” is very important here, so as to make clear that there is only a certain analogy with the mystery of the Incarnation.
- Conference dated February 5, 1941, in KMK p. 428
Note: this citation was taken in @Geremia's answer in the CSE link provided above.