It must be that the place where Moshe was placed was a pubic bathing place off the Nile (a canal?), which was safe from crocodiles, and it was open both to commoners and to the royal family though at different times. For instance it could be that in mid-day it was open to the public, and during this time period, Moshe's mother brought him and placed him in the reeds. At this same time, Moshe's sister also hid herself in the area, 2:4, as how else could she have gotten so close to the princess to speak to her, 2:7. After the commoners were cleared out, then the daughter of Pharaoh came to bath, and found Moshe. 2:6 records that Moshe was crying, and it is possible that he had been crying for a while, and this is what led Pharaoh's daughter to find him.
Thus, Moshe being found in the basket was most likely not a coincidence, but Moshe's mother placed him in a place where she knew that Pharaoh's daughter was known to bath. It is even possible that Moshe's sister would have taken Moshe out of the basket, if Pharaoh's daughter did not find him since he could only have survived a short while in the basket. Accordingly, Moshe's mother did not abandon Moshe.
It has been pointed that Moshe's early life follows the literary pattern of the birth of heroes (Fox, 1995, p. 260, see also Raglan, 1936). This pattern includes that the father is a relative of the mother, there is an attempt to kill the child at birth, the child is abandoned at birth, foster parents raise the child, and eventually the child becomes a hero. According to this idea, maybe the Torah recorded how Moshe's early years followed this pattern in order for people to be more inclined to accept Moshe. However, Sarna (1991, pp. 267,268) points out that there are significant differences between Moshe and the hero pattern. For instance, usually the child is abandoned to die while here Moshe was not abandoned at all. Also, usually the child was born into royalty and then when abandoned grew up in humble circumstances, while here Moshe's real parents were slaves and the daughter of the king adopted and raised him.
Instead, maybe the point of this episode is to show that Moshe was a regular human being. Moshe was about to perform many miracles, which might lead one to think that Moshe was almost like a god. Thus, the Torah records a normal birth for Moshe that he did not just appear, and we have an image of Moshe as a baby, crying, which gives us a depiction of Moshe as a regular baby as opposed to always thinking of Moshe as a man who does miracles. Furthermore, we see how precarious his life was when he was a baby, and this precariousness also makes him more human.
This idea suggests a reason why the Torah records Moshe’s birth and not Avraham’s. We are not told Avraham's early childhood since he did not perform miracles, but by Moshe, with his incomparable greatness, the Torah wants to point out from the very beginning that he was a human being and not divine.
Fox, Everett, 1995, The Five Books of Moses: A new translation, New York: Schocken Books.
Raglan, Lord (Fitzroy Richard Somerset), 1936, The Hero: A study in tradition, myth and drama, London: Methuen & Co., Ltd.
Sarna, Nahum (1923-2005), 1991, The JPS Torah Commentary: Exodus, Philadelphia: The Jewish Publication Society.