In most cases, when a saint is recognized in the Catholic Church, their feast day is assigned to the day of their death. This day would technically be the anniversary of their entrance into Heaven.
In other words it would be their "heavenly birthday."
However, this can get complicated, as the Church has canonized thousands of saints over the centuries and it is impossible to assign each saint on the liturgical calendar to the precise date of their death.
St. Thérèse's feast day
If St. Thérèse were to be celebrated on her date of death, her feast day would be September 30.
The 2004 Roman Martyrology mentions her on this day:
At Lisieux in France, anniversary of the death of Saint Thérèse of the Child Jesus, whose memorial is celebrated tomorrow.
The primary problem with celebrating St. Thérèse on September 30 is that it is also the feast day 0f St. Jerome on the universal calendar of the Catholic Church.
In order for St. Thérèse to be celebrated on September 30, the Church would have had to suppress St. Jerome's feast, which would have been difficult, as he is recognized as a Doctor of the Church.
The next closest day was October 1, but initially this was not possible, as it was the feast of St. Remigius, who is the "Apostle of the Franks." This was a major feast in France at the time, as he is a principal patron of France.
October 2 was taken by the feast of the Guardian Angels, and so initially her feast was celebrated on October 3, which has no connection to her life, but was the closest "open" day in the Church's universal calendar.
By 1969 St. Thérèse was becoming one of the most popular saints of modern times, and when the liturgical calendar was reordered, St. Remigius (who was not as well-known in the universal Church) was removed from the general calendar and St. Thérèse took his place on October 1.
This was the closest the Church could get to September 30, the actual date of her death.
Some say that October 1 was her first "full" day in Heaven, though technically there are no "days" in Heaven.
The Church simply wanted to honor St. Thérèse with a day that made the most sense and which respected other important days on the calendar.