Mel claims she looked into her flames searching for Arya and saw only once a vision:
A grey girl on a dying horse, travelling in haste, along the western shore of a gigantic lake. In a surrounding area of Hills, Trees, fields, and stones. The girl is avoiding villages that are also in the area.
A girl does arrive at the Wall and it is believed that Mel’s vision was wrong. Alys Karstark is the grey girl travelling along the western shore of Long Lake heading to the Wall for Jon’s protection from her unwanted marriage.
Alys is a red herring though.
Mel admits only to herself that she has no clue when this moment happens or where this girl really is. So how can she know the girl is heading North? Mance only suggests Long Lake based off this assumed "heading North to the Wall" idea he got from Mel.
*Mance thinks the girl is clever for avoiding the dangerous King's Road.
Early on Arya learns this lesson from Yoren:
"We're not far from Gods Eye," the black brother said one morning. "The kingsroad won't be safe till we're across the Trident. So we'll come up around the lake along the western shore, they're not like to look for us there." - Arya, ACoK
Arya ends up doing just that as she leads Gendry and Hotpie. Arya then travels with the The Hound and has this same lesson drilled into her later on.
*Mel says the girl is avoiding villages.
So she stayed with the Hound.
They rode every day, never sleeping twice in the same place, avoiding towns and villages and castles as best they could. – Arya, ASoS
Every day they marched, and every night she said her names, until finally the trees thinned and gave way to a patchwork landscape of rolling hills, meandering streams, and sunlit fields, where the husks of burnt holdfasts (villages) thrust up black as rotten teeth. It was another long day's march before they glimpsed the towers of Harrenhal in the distance, hard beside the blue waters of the lake. – Arya, ACoK
~*~
The setting sun made the tranquil surface of the water shimmer like a sheet of beaten copper. It was the biggest lake she had ever seen, with no hint of a far shore. — ARYA IV, A CLASH OF KINGS
~*~
To the east, Gods Eye was a sheet of sun-hammered blue that filled half the world. Some days, as they made their slow way up the muddy shore (Gendry wanted no part of any roads, and even Hot Pie and Lommy saw the sense in that), **Arya felt as though the lake were calling her. She wanted to leap into those placid blue waters, to feel clean again, to swim and splash and bask in the sun. — ARYA V, ACoK
”And what else do you know?" It is snowing in the riverlands, in Westeros, she almost said. But he would have asked her how she knew that, and she did not think that he would like her answer. - The Blind Girl, ADwD
So what will the Gods Eye look like when Arya returns to Westeros?
Mel describes the surroundings of this grey girl and they fit the area around the God's Eye lake.
“Hills. Fields. Trees. A deer, once. Stones. She is staying well away from villages. When she can she rides along the bed of little streams, to throw hunters off her trail.”
Recall Arya's chase on horseback and the terrain George describes after she becomes a captive and tries to escape:
"Why do I have to see Lord Beric?" she asked quietly.
"We bring him all our highborn captives," said Anguy.
Captive. Arya took a breath to still her soul. Calm as still water. She glanced at the outlaws on their horses, and turned her horse's head. Now, quick as a snake, she thought, as she slammed her heels into the courser's flank. Right between Greenbeard and Jack-Be-Lucky she flew, and caught one glimpse of Gendry's startled face as his mare moved out of her way. And then she was in the open field, and running.
North or south, east or west, that made no matter now.
Made no matter indeed! The passage is long but Arya races over Hills, trees, fields, and stones.
If the Lake in the vision is the Gods Eye, then that really narrows things down: Jeyne wouldn't be riding alone anywhere near the Gods Eye. Alys is the obvious red herring.
What are the chances that the grey girl in Mel's vision is Arya, at a future point returned to Westeros, travelling South along the Western shore possibly heading somewhere important… like the Isle of Faces?
Arya felt as though the (Gods Eye) lake were calling her. She wanted to leap into those placid blue waters, to feel clean again, to swim and splash and bask in the sun. But she dare not take off her clothes where the others could see...
From up here, she could see a small wooded island off to the northeast. (The Isle of Faces, Holiest place for the Children of the Forest) Thirty yards from shore, three black swans were gliding over the water, so serene . . . no one had told them that war had come, and they cared nothing for burning towns and butchered men. She stared at them with yearning. Part of her wanted to be a swan the other part wanted to eat one. - (Arya, A Clash of Kings)
While travelling with the Hound near the Trident, Arya marks 4 trees with her name and prays for a Red priest to find her in their flames. Mel desperately was trying. Perhaps she has succeeded, but got the place and time completely wrong? She often misinterprets things. The grey girl (Stark) travelling on a dying horse (signifying a long journey or a quick escape leading to a wounded horse) heading South along the Gods Eye, not Long Lake in the North as Mel wanted to believe was true so she can win Jon’s trust.