Did Ned Stark raise his children properly?

Is that what Jon's name should have been?

Bastards are named not for their place of birth, but for theirs place of birth .

Remember, Ned claims Jon is his own child. While he was born outside the north, he was brought north by a north man. It has more to do with who the bastard was born to and where they live than where they are actually born.

Explanation of the bastard naming:

Each of the nine constituent regions of the seven kingdoms has bastard surnames ordained by custom and not by law. Bastards with a highborn parent are given these last names to match them from their fathers' homes . Parents can give a bastard a different last name if they wish, e.g. B. Tyrion Tanner. Bastards with no known relationship with a noble house do not have a surname like other little people.

Examples of naming hybrids:

The last name a bastard received seems connected to the place to be on which the child is raised, although this is not a single rule. Because of this, bastards who are half-siblings can have different last names. For example, King Robert I Baratheon's oldest bastard born in the valley is called Mya Stone, while his Stormland bastard (fathered by a noble out of reach) is called Edric Storm. While Aegor Rivers grew up in Stone Hedge in the Riverlands, his half-brother Brynden was also called Rivers, although he had spent at least the first few years of his life with his mother at King's Landing in the Crownlands.

So it should have depended on which parent took custody of Jon.

If it had been Rhaegar, Jon would probably have been a water since Rhaegar would have brought him back to the Crownlands.

If it had been Lyanna, Jon would probably still have been a snow since he would have been brought back north.