The passuk tells us that Moshe was born, and Yocheved hid him for three months. The Gemara says that she gave birth three months before the Egyptians guessed, because she had gotten pregnant three months before she had gotten remarried to Amram, when was before she had received her divorce from him. The Egyptians counted nine months from the remarriage, not realizing she was pregnant from before. Thus she was able to keep him hidden in her home for three months. At that point she had no choice but to put him in the river, where he would be found by Batya.
Rabbenu Bachai points out that Moshe was born on the seventh of Adar. If we count three months later, we land on the sixth of Sivan, which is Shavuos. He says that the day would be a day of distress and salvation for Moshe. It was a day of distress because his life was in jeopardy as he was placed in the river, and a day of salvation, for on it he would receive the Torah.
Perhaps this hints to the possibility that Shavuos has the potential to be a significant day in the Geulah process.