The lions in Santiago's dreams are the ones that he observed as a young boy sailing on large ships. He remembers this time of his life as a pure pleasure, with no negative memories attached to it, and becomes happy whenever he dreams of the lions playing on the beach:
He no longer dreamed of storms, nor of women, nor of great occurrences, nor of great fish, nor fights, nor contests of strength, nor of his wife. He only dreamed of places now and of the lions on the beach. They played like young cats in the dusk and he loved them...
(Hemingway, The Old Man and the Sea, Google Books)
One interesting comparison is the mention of Santiago's wife; she died, and he has removed her picture from his shelf because it makes him sad. Despite the many happy memories he must have of her, the sad memory of her death makes him shunt those memories aside, so he doesn't have to think of them. Because all the memories of the lions are good, he can dream of them and remember his youth when he had no fear, no guilt, and no sorrow.
One potential meaning behind the dream of the lions is connected to an idea of the afterlife or heaven. An old man, Santiago no longer dreams of success and strife in life. Instead, he dreams of a different "place" altogether, somewhere outside of the social context of achievement, success and failure that he has lived within for so long.
Santiago's narration recounts the idea that the lions he had seen on the coast of Africa are part of a picturesque and pristine set of images and memories. The sights and smells are calm and lovely to him.
His Africa is a place of peace.
Notably, the many events and elements of his life and his life's ambition are absent from the dream of the lions.
"He no longer dreamed of storms, nor or women, nor of great occurrences, nor of great fish, nor fights, nor contests of strength, nor of his wife. [...] only of the lions on the beach."
After a lifetime of these things, Santiago, El Campeon, now dreams of an opposite set of images. The "white peaks" of the Canary Islands with their playful lion cubs are foreign to his experience - a very different setting from the rather masculine and codified world he has lived in.
Santiago is one of Hemingway's most sentimental characters, but at the same time adheres to a code of behavior that leads him to revel in his memories of arm-wrestling and in his admiration for Joe DiMaggio. This is all left behind when he dreams of the lions playing in what amounts to a paradise.
Santiago is religious. He prays and reflects on the relationships of man to the larger world in ways that are consistent with a mystical/Catholic sensibility. That his heaven would be a place near the sea in a foreign land characterized by natural beauty is not surprising.
So, when Santiago lies utterly exhausted in his ambiguous victory/defeat at the end of the narrative, he dreams of the lions - perhaps signifying that he has come to the end of his life and moved on.
"Rather than a mere triumph over nature, he has, with great dignity and humility, achieved atonement (at-one-ment), oneness with nature" (eNotes).
We might take a moment also to mention that the religious imagery of the text can be connected to the lions. Christian imagery is used throughout the narrative and Hemingway's The Old Man and the Sea is routinely interpreted as a religious allegory of sorts, wherein the old man suffers but acts on his nature, accepting his fate and submitting to the will of Nature (or God).
The lion is used as a symbol for Christ in Christian iconography. We might argue then that the dreams of the lions are not only a dream of paradise generally but specifically a dream of encountering Christ in the afterlife or something along those lines.