Why does Beowulf ask Wiglaf to build him a tomb and call it Beowulf's tower?
Beowulf asks Wiglaf to build him a tower as a tomb to signify that he, Beowulf, was the greatest warrior of all time.
He was still himself,
Alive, aware, and in spite of his weakness
He had many requests. He wanted me to greet you
And order the building of a barrow that would crown
The site of his pyre, serve as his memorial,
In a commanding position, since of all men
To have lived and thrived and lorded it on earth
His worth and due as a warrior were the greatest.
Beowulf's life was marked by feats of heroism. Even in death, he was a hero—defeating a dangerous dragon that his own warriors ran from. His tower will serve as a legacy to his memory, even as his nation is taken over by outsiders.
Beowulf wasn't only the king of his people. His strength and reputation as a warrior kept the Geats safe from invasion by foreign people. They're aware of their own vulnerability at his passing, which contributes to their grief.
They do build the tomb. The poem says:
Then the Geat people began to construct
A mound on a headland, high and imposing,
A marker that sailors could see from far away,
And in ten days they had done the work.
It was their hero’s memorial; what remained from fire
They housed inside it, behind a wall
As worthy of him as their workmanship could make it.
And they buried torques in the barrow, and jewels
And a trove of such things as trespassing men
Had once dared to drag from the hoard.
They let the ground keep that ancestral treasure,
Gold under gravel, gone to earth,
As useless to men now as it ever was
Beowulf is gone, but his legacy remains and will be seen by everyone who passes his resting place. His people celebrate him at his tomb after they finish building the tower—but the celebrations and tales of his bravery are marked with weeping.
Beowulf wants the tall tower built at the edge of the water so that sailors can see it from a distance and remember his name. Thus Beowulf wishes to be remembered after his death. Also, as a man of the sea, he identifies with those who will come in their own journeys over the sea after his death. He wants the tower built so that "boats in the darkness / And mist, crossing the sea, will know it."
As recounted in "The Burning of Beowulf's Body," Beowulf's request is honored. The Geats build the tower "strong and tall," working on it for "ten long days." After Beowulf's monument is completed, the Geats seal his ashes within its walls and also bury the treasures won by Beowulf and Wiglaf. Once Beowulf's burial is complete, "twelve of the bravest Geats / Rode their horses around the tower, / Telling their sorrow . . . ."
In addition to the previous response, by burying the treasure within the tower a statement is being made about the cowardice of the men (besides Wiglaf) who flee rather than face the dragon. The king was expected to reward those who were loyal to him and, as Wiglaf points out to the men, their flight has disgraced them and their families. There will be no treasure to share. The tower becomes not only a monument to the greatness of Beowulf but also a reminder of the failure of his soldiers.
Remember too that to the people of this time, Fame is viewed as the thing most desired - the only thing that lasts beyond death. So this monument tower would also keep Beowulf's name and story in front of the people.