Hagrid's day started off normally, he had gone into the Forbidden Forest in the morning to ensure the thestrals were fed and doing well. While he was walking back to his hut a silvery phoenix had appeared and then spoke, the silvery apparition asked Hagrid to go speak to Headmaster Dumbledore as soon as he could.
Professor Dumbledore asked him to please deliver young Harry Potter's school letter because all the previously sent ones had gone unanswered. Hagrid of course agreed, he had no problem helping Professor Dumbledore with such a task, and he would gladly see Harry again.
After their discussion Hagrid headed out. He wondered for a moment "How am I gonna get to" Squints eye at letter "Hut on a rock? Now where is that?". He decided he would use a Thestral to make the trip, and that hopefully the incredible sense of direction it had would aid him in finding his destination.
Through some miracle of magic the Thestral had somehow known where to fly and Hagrid found himself approaching a small rocky outcrop off in the sea. On the small island he spotted a small hut. While at first he found it strange, what with the stormy weather it did not look like a nice place to be at all. He realized however that its not like muggles could control or even predict the weather, they likely had planned a vacation here since it was summer. Seems Shoddy British weather had ruined that plan.
Hagrid brought the thestral in for a landing. Due to the story weather it was currently quite dark and hard to see but they still managed well enough. As Hagrid approached the door of the hut he looked around, he then spotted a dark and short figure on the far side of the island facing out towards deeper sea.
As Hagrid approached the small figure became clearer, it was a young boy with Black hair. The child glanced back over his shoulder at Hagrid and revealed his shockingly bright green eyes. "Harry, Harry Potter?" Hagrid spoke, but the child turned back to face the sea and spoke "What now I don't have time for this, I already said I would not marry the Fae princess". "the Fae what now-" but again Hagrid was interrupted, "I told you not now, I'm busy" Harry said "it's here". Hagrid experienced a moment of confusion before seeing what Harry was looking at. A few dozen meters out in the sea a dark splotch had appeared. Then it erupted and a sea serpent appeared out in front of them. It started to approach the island making it apparent just how large it was. 10 meters worth of the sea serpent was out of the water, making it tower over even Hagrid.
Hagrid stared on in awe while Harry just sighed, and muttered "a sea serpent then, I don't know what I expected, wouldn't have guessed any lived around here, but it's just my luck that they do". Hagrid then helpfully pointed out "Actually this specific guy here is a North Atlantic Sea Serpent, they live out in the deep ocean in the North Atlantic and are never seen near the coasts, in fact none are known to live in the North Sea at all". Hagrids mind then caught up to him and he started to panic, as he reached down for his umbrella he called out "Harry you have to run, it shouldn't be able to get into the hut, I'll protect you. Before Hagrid had even said the word run the Sea Serpents throat had started to bulge, it then opened its mouth releasing a pressurized jet of water at the two. Harry calmly raised his hands together then pulled them apart causing the water jet to split in two and pass them by. Harry then looked up into the stormy sky and smirked, he then looked back down at the serpent and called out "You have one chance, leave us in peace or I will respond in kind". The serpent released a sound that was surprisingly high pitched for something of its size and then reared back as if to prepare to strike forward. "So be it " Harry whispered as he raised his hands to the sky. Meanwhile the sky had continued to darken and rain had begun to fall, the storm was intensifying. "By the power of this storm I bid you Begone!" Harry yelled out while throwing his hands forward and down. The sky flashed, a lightning bolt fell from the heavens and struck the serpent with a deafening roll of thunder. It spasmed and then fell still, Hagrid and Harry were still blinded by the lightning, but they heard the crash as the Sea Serpent toppled back into the sea.
And that is the story of how Hagrid properly met the unusual young boy who is Harry Potter.