This can depend. In the Old Jedi Order, as Jedi were trained from infancy, they constructed their own lightsaber while as a Padawan learner under a Master. Padawans are generally chosen around age thirteen and become Knights in their twenties, so the teenage years for them.
In Luke Skywalker's New Jedi Order, many of his first students were already full adults and did not construct a lightsaber until much later: Gantoris, Kyp Durron, Corran Horn, Streen, and others do not construct one until they are much older, with Corran even approaching thirty years of age when he constructed his, and Streen even being an old man by the time he is Knighted.
The next generation of Skywalker's Jedi, like his niece and nephews Jaina, Jacen, and Anakin, construct theirs when they are fourteen years old.
By the time Skywalker's own son, Ben, is fourteen, he also constructs his, and it is suggested that such construction is indicative of ascension to Knighthood; though Ben is young for a typical Knight.
So it seems that the answer would probably be fourteen, fifteen years old, probably matching the period in which the person will no longer physically grow so the lightsaber well matches the needs they have in grip length and whatnot, as the Jedi will potentially carry that same lightsaber for the rest of their lives.