Sorting Harry Into A House

Question

When the sorting hat was placed on Ronald Weasley's head, it instantly decided Gryffindor, just because he was another Weasley.

On the other hand, the Black family were all Slytherins, except Sirius.
I'm guessing Sirius pleaded to be a Gryffindor.

If Harry pleaded not to be a Slytherin, why didn't the sorting hat go:

Not Slytherin... OK, that's easy to arrange. Ah, another Potter. Gryffindor!

The sorting hat thought for a while.
At one point, it considered sorting Harry into Slytherin.

Maybe to increase the suspense.

Answer 1

There are 3 issues with the assumptions in the question, which should explain away the confusion when addressed:

1. The hat does not always decide quickly (as it would if it was merely ancestry based).

   Harry wasn't the lone exception for deliberations:

   Sometimes, Harry noticed, the hat shouted out the house at once, but at others it took a little while to decide. 'Finnigan, Seamus', the sandy-haired boy next to Harry in the line, sat on the stool for almost a whole minute before the hat declared him a Gryffindor.

   ...

   When Neville Longbottom, the boy who kept losing his toad, was called, he fell over on his way to the stool. The hat took a long time to decide with Neville. When it finally shouted 'GRYFFINDOR'...

   Also, covered in this SFF question or this.

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2. Ron being placed into Gryffindor wasn't about being a Weasley (the Hat was basically jesting) - there are examples of people NOT being placed in same house as their families (Sirius Black, as you yourself noted. Or Andromeda Tonks. Or Patil sisters who got split between houses):

   “Oh no, not necessarily,” said Hermione. “Parvati Patil’s twin’s in Ravenclaw, and they’re identical. You’d think they’d be together, wouldn’t you?” (GoF, Ch 12, The Triwizard Tournament)

   The hat's main reason for placing you in the house is how well suited your potential is to the house.

   Ron - with his straightforwardness and bravery - would be a no-brainer for Gryffindor.

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3. I'm not sure where you got the idea that the Hat - after Harry asked not to be in Slytherin - dithered for a while about other houses. It considered other houses before Harry stated his "not Slytherin" preference:

   'Hmm,' said a small voice in his ear. 'Difficult. Very difficult. Plenty of courage, I see {{ Gryffindor - DVK }}. Not a bad mind, either {{ Ravenclaw - the Hat was clearly off its rocker there - DVK }}. There's talent, oh my goodness, yes – and a nice thirst to prove yourself, now that's interesting... So where shall I put you?'

   ... and placed him in Gryffindor right away after rejecting Slytherin by his request:

   Harry gripped the edges of the stool and thought, 'Not Slytherin, not Slytherin.'

   'Not Slytherin, eh?' said the small voice. 'Are you sure? You could be great, you know, it's all here in your head, and Slytherin will help you on the way to greatness, no doubt about that – no? Well, if you're sure – better be GRYFFINDOR!'

All unattributed quotes are from "Harry Potter and Philosopher's Stone", Chapter 7, The Sorting Hat

Answer 2

Harry and Dumbledore talk about it briefly at the end of Chamber of Secrets:

"So I should be in Slytherin," Harry said, looking desperately into Dumbledore's face. "The Sorting Hat could see Slytherin's power in me, and it -"

"Put you in Gryffindor," said Dumbledore calmly. "Listen to me, Harry. You happen to have many qualities Salazar Slytherin prized in his hand-picked students. His own very rare gift, Parseltongue - resourcefulness - determination - a certain disregard for rules," he added, his mustache quivering again. "Yet the Sorting Hat placed you in Gryffindor. You know why that was. Think."

"It only put me in Gryffindor," said Harry in a defeated voice, "because I asked not to go in Slytherin..."

(source: Harry Potter and the Chamber of Secrets, chapter eighteen, Dobby's Reward)

So it seems the Hat was very strongly inclined to place Harry in Slytherin, but Harry's intense desire to avoid that fate swayed it into placing him in Gryffindor instead.

Answer 3

The initial confusion, where the Hat briefly considers placing Harry in each of the four houses, is pretty easy to understand: Harry does embody the characteristics of all four houses, and it's hard at a glance to determine which quality is more significant (benefit of being the protagonist, I guess). But I assume your question is more about the second part, after Harry asks not to be put in Slytherin and the Hat double-checks.

With the benefit of hindsight, this seems to be a hint at the nature of Harry and Voldemort's relationship. The point of the Hat is that it determines the kind of person you are, though it's never been made clear exactly how that works¹.

Whatever the method, it's not inconceivable that the Hat would have detected the connection between Harry and Voldemort. From the Hat's perspective, this is a bit of a conundrum: you have the textbook Slytherin, practically the reincarnation of Salazar, and the quintessential Gryffindor, both in the same body. What do you do when you flip a coin and it falls on its edge?

Harry solves this problem for the Hat by explicitly wanting to be in Gryffindor, but the Hat still isn't sure; what we know as Voldemort's influence would make a really good SLytherin candidate. So it does what anyone would do: it asks.

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¹ In fact the Hat seems to be able to determine the kind of person you are even if you don't know yourself, as in the cases of Peter Pettigrew (he had his moment of bravery at the end) and Neville Longbottom

Answer 4

It did. It Sorted Harry to the house that his parents had been in -- Gryffindor.