St. John begins to tell us of the discovery of Jesus’s resurrection:

Early on the first day of the week, while it was still dark, Mary Magdalene came to the tomb and saw that the stone had been removed from the tomb. So she ran and went to Simon Peter and the other disciple, the one whom Jesus loved, and said to them, “They have taken the Lord out of the tomb, and we do not know where they have laid him.” Then Peter and the other disciple set out and went toward the tomb. The two were running together, but the other disciple outran Peter and reached the tomb first. He bent down to look in and saw the linen wrappings lying there, but he did not go in. Then Simon Peter came, following him, and went into the tomb. He saw the linen wrappings lying there, and the cloth that had been on Jesus’ head, not lying with the linen wrappings but rolled up in a place by itself. Then the other disciple, who reached the tomb first, also went in, and he saw and believed; for as yet they did not understand the scripture, that he must rise from the dead. Then the disciples returned to their homes.

It is not quite clear what the other disciple believed, given that “as yet they did not understand the scripture, that he must rise from the dead.” St. Augustine interprets this to mean that they believed what Mary had said:

Here some, by not giving due attention, suppose that John believed that Jesus had risen again; but there is no indication of this from the words that follow. For what does he mean by immediately adding, For as yet they knew not the scripture, that He must rise again from the dead? He could not then have believed that He had risen again, when he did not know that it behooved Him to rise again. What then did he see? What was it that he believed? What but this, that he saw the sepulchre empty, and believed what the woman had said, that He had been taken away from the tomb?

In any case, whether it was this or something else, St. John continues:

But Mary stood weeping outside the tomb. As she wept, she bent over to look into the tomb; and she saw two angels in white, sitting where the body of Jesus had been lying, one at the head and the other at the feet. They said to her, “Woman, why are you weeping?” She said to them, “They have taken away my Lord, and I do not know where they have laid him.” When she had said this, she turned around and saw Jesus standing there, but she did not know that it was Jesus. Jesus said to her, “Woman, why are you weeping? Whom are you looking for?” Supposing him to be the gardener, she said to him, “Sir, if you have carried him away, tell me where you have laid him, and I will take him away.” Jesus said to her, “Mary!” She turned and said to him in Hebrew, “Rabbouni!” (which means Teacher). Jesus said to her, “Do not hold on to me, because I have not yet ascended to the Father. But go to my brothers and say to them, ‘I am ascending to my Father and your Father, to my God and your God.’” Mary Magdalene went and announced to the disciples, “I have seen the Lord”; and she told them that he had said these things to her.

St. Augustine comments on the fact that Mary stayed at the tomb while the others returned home:

“But Mary stood without at the sepulchre weeping.” For while the men returned, the weaker sex was fastened to the place by a stronger affection. And the eyes, which had sought the Lord and had not found Him, had now nothing else to do but weep, deeper in their sorrow that He had been taken away from the sepulchre than that He had been slain on the tree; seeing that in the case even of such a Master, when His living presence was withdrawn from their eyes, His remembrance also had ceased to remain. Such grief, therefore, now kept the woman at the sepulchre. And as she wept, she stooped down, and looked into the sepulchre. Why she did so I know not. For she was not ignorant that He whom she sought was no longer there, since she had herself also carried word to the disciples that He had been taken from thence; while they, too, had come to the sepulchre, and had sought the Lord’s body, not merely by looking, but also by entering, and had not found it. What then does it mean, that, as she wept, she stooped down, and looked again into the sepulchre? Was it that her grief was so excessive that she hardly thought she could believe either their eyes or her own? Or was it rather by some divine impulse that her mind led her to look within?

While St. Augustine finds it curious that Mary continued to look, he gives the explanation himself when he says that she “was fastened to that place by a stronger affection.” If we are looking for something, we will look more and harder to the degree that we care about it more. If we lose something and care about it a lot, we might very well search the same places repeatedly, even multiple times. Of course, this is usually because we think we might have missed it, but sometimes we even search again in places where there is no realistic possibility of having missed it. And in the case of Mary Magdalene, she could believe it possible that she missed some remaining clue. In any case, the very fact that she cared more than the others explains her behavior sufficiently; there is no need to rationalize every aspect of it.

In this account, Mary is the one who announced to the other disciples, “I have seen the Lord.” The Ambrosian rite describes this,

O worthy conversion, that merited to receive so great a gift, that she who was formerly deservedly held fast in the jaws of the ancient dragon, now rejoicing in complete freedom, should merit to be the first to announce the Lord’s Resurrection to the Holy Apostles.

In this way a close connection is made between Mary’s love and the fact that she was the first to recognize the resurrection of Christ. One who cares more about something, is more likely to find it.