Author's Note

Would Harrenhal be a good new home for a upstarting Lord in Westeros?

In its prime? Absolutely

It is in the top 5 greatest Castles in Westeros (#4 actually, being beaten by Casterly Rock, Storm's End, and the Eyrie) but it is in a bad position, economically.

Harrenhal is close to the Trident and could easily reach two of the main regions via the Trident (The Vale and North), and was attached to the Blackwater via the God's Eye, and thus the Reach, Stormlands and open sea. Overall, it's a pretty good connection.

However, not anymore

Only 60% of its towers are destroyed and it is rather uncomfortable. It is also far away from the Sea and Dragonstone, both are two incredible things for the Targaryens. Not only that, it is a great sign for the Targaryens. If anyone tries to fight them, they'll suffer the same fate of Harrenhal. The powerful moment would be undercut if they moved in.

though not anymore because they are recently dethroned.

King's Landing is in a great spot, located near the sea and Dragonstone. It is a major part of Western trade routes, arguably the most important. Furthermore, cities are very important to produce armies. House Hightower, Manderly, and Lannisters are the most powerful because they have cities.

Cities only form in important spots, and Harrenhal isn't in important spot, while King's Landing is.

cons:

The castle of Harrenhal is the ultimate symbol of hubris, and even its name is that of a defeated king (Harren the Black). It also has a reputation for bad luck. This is not a good starting point for a new king building a new capital.

Building an entire city from scratch is no easy task. It would require a huge amount of expense and a considerable amount of time.

Harrenhal is in the interior. All the current cities of Westeros are ports, because ships are the fastest mode of transport in such a huge country (with pre-modern forms of transport). A city in the interior would require a whole lot of new infrastructure to connect it to the rest of the country.

Harrenhal has few natural defences. It's on the shore of the God's Eye, but a lake isn't much of a defence - armies can easily go around it. King's Landing, on the other hand, is protected on two sides, by the sea and the Blackwater Rush.

Pros:

Harrenhal is in the geographic centre of the Seven Kingdoms. It's in the middle of the Riverlands, roughly equidistant from the borders of the Westerlands, the Reach and the Vale, and of course, far closer to the North than King's Landing is. It's a good location for a capital.

The castle, despite its unfortunate symbolism (mentioned above) is also enormous and easily capable of housing the royal court. It would require considerable repair and rebuilding (the melted towers would definitely have to be pulled down), but on the whole, from the practical point of view, it would make a good residence.

Harrenhal is not too far from the Kingsroad, and from its junction with the Riverroad and the Highroad. A short branch of the Kingsroad would enable Harrenhal to be connnected by road to a large part of the country.

Harrenhal is also not very far from the Trident, which is a navigable river. A new port built on the Trident at its closest approach to Harrenhal would increase Harrenhal's connectivity greatly.

Harrenhal is near the Isle of Faces, which is a holy site for the followers of the Old Gods. If the new capital city promoted itself as a pilgrimage site, it would grow quickly. but I don't think so.

But I personally think it's a good place for big family.