When Saber died, she should have lost both Excalibur and Avalon. The Einzbern are able to locate the latter in the present time, which they used as catalyst to summon her. Why is it that Saber still had Excalibur on her person when she was summoned as HS? (IIRC, it is sheathed by invisible air). The sword itself should have been left behind in the real world like Avalon was.
Fate/side material (2004-1-30), p.056 defines Heroic Spirits as:
Heroes who achieved great things in life, and became objects of worship after death.
Saber is actually a special case in this matter. Saber did not die. When Saber was on the brink of death she made a contract with the world to become a heroic spirit after her death. Since the stories we see are in a time much later than the time Saber is in, she can of course be summoned, since Heroic Spirits are timeless. However, at the time of making this contract, she was not in possession of Avalon.
From Fate/complete material III World material:
In the civil war that raged at the end of Artoria's reign, she and her son Mordred mortally wounded each other. At that moment, she made a pact with the World to serve as a Heroic Spirit after death. In exchange, she wished that the Holy Grail would redo the selection of the king.
Saber had lost Avalon before her death, and its loss had been engrained into the legend of King Arthur. While she did have Bedivere throw Excalibur back to the Lady of the Lake, it was with her until soon before she would pass away. It was then that Saber made the contract with the world, requesting she be given an opportunity to seek the grail in exchange for her becoming a Heroic Spirit.
Also from Fate/complete material III World material:
In legend, the loss of this sheath cast an ominous pall over the life of King Arthur, eventually cascading into the collapse of the kingdom. In modern times, the Einzberns excavated Avalon at Cornwall and passed it into the hands of Emiya Kiritsugu.