Mid–summer falls on the vernal equinox. Britain celebrates this on June 24th. Since Celtic days begin at sundown with the dark or night half of the day, followed by the light or day half, ending at the next sundown, mid–summer's eve begins at sunset on June 23rd. The Celts will traditionally associate mid–summer's eve with the færie; therefore, this will be an auspicious time for Gandolf's mother to remove her child to safekeeping.
This dating will mean that Gandolf is about seven weeks old at this time. I have wrestled with the timing and conclude that Gandolf, here, refers to human time, since his mother will be mortal. If Gandolf were referring to wizard time, he would probably mean the mid–summer's eve before which would work out to be about forty–five weeks. This would put Gandolf much older, seven weeks shy of a year, and I would wonder whether his mother could successfully shield her second child's existence from common folk for such a lengthy period of time.