Why organize the meeting here?

Question

In the finale of season 7 of Game of Thrones,

There is a meeting between Lannisters, Starks and Targaryens for a truce at the Dragonpit.

Who decided the meeting place and why?

Answer 1

Cersei probably decide to organize the meeting in Dragonpit for its background

We don't know for sure who decided what for the meeting, but we can speculate that Tyrion organized the meeting, but let Cersei decide the place. And the place is not neutral.

Dragonpit is a good place to show the decadence of Targaryen dynasty

Let's see briefly the history of the Dragonpit :

Rhaeny's Hill before Dragonpit, and destruction of the Sept of Remembrance :

Dragonpit is located at Hill of Rhaenys, named after the sister-Queen of Aegon I. Aegon was really attached to Rhaenys, who died during the first Dornish war. In memory of her, Aegon built the Sept of Remembrance

The Sept of remembrance was an important place during the Faith Militant uprising, a huge rebellion against the Targaryens. Long story short, Maegor the cruel burnt the sept to the ground. He was the one who decided to built the Dragonpit.

Dragonpit's construction

Because Maegor had killed all those who had built the Red Keep, to hide its secrets, many fled rather than work on the Dragonpit. Maegor had to use city prisoners, with supervisors from Myr and Volantis. We are starting to understand where the epithet of Maegor come from.

Dragonpit and size of dragons

We know that the last dragons were really small. The reasons seems to be Dragonpit : Dragons which are locked up, e.g. the Targaryen dragons housed in the Dragonpit , did not grow as large as dragons left free to fly as they wished.

Dragonpit's destruction

Dragonpit was destroyed in 130AC, during riots, in the middle of the Dance of the Dragons. The Shepherd, a kind of prophet, lead the smallfolk to Dragonpit, to kill the dragons. 4 dragons died, and the last one, Dreamfyre flew into the pit's great dome above, which cracked on impact and half of it came tumbling down, crushing her.

So, to resume it, Dragonpit was constructed by a mad king after a rebellion, and destroyed during riots in context of civil war, and lead to the exctinction of dragons.

There are other reason for this place :

• It is one of the most isolated place in King's Landing, allowing them be isolated from smallfolk and armies
• Out of universe, Dragonpit is one of the most important parts of King's Landing that have been never seen before. Always good to show new places!

Answer 2

All of Them

Let's look at it from each characters perspective first:

• Cersei - She is controlling and the current Queen. If she is to meet with them she is going to do it on her terms and that is at Kings Landing. She holds power at Kings Landing and is more protected there. She also probably chose the Dragonpit for the reason Dany says herself "This place was the end for my family." It is to show Dany that even great dragons can become small and weak and be overtaken by the next generation.
• Jon - He needs the support of everyone he can if he is to win the war over the undead. If he invites Cersei to him she will almost certainly decline suspecting it is a trap. So he takes the evidence to her which makes complete sense as it will also make her trust him a little bit more.
• Dany - She knows if she asks Cersei to go to her she will say no, she knows if she doesn't go to Cersei she won't help as the two don't trust each other at all. But Dany going to Cersei will at least show a bit of trust between them.

Also, as @Skooba, says:

It was a pretty good place to compromise on IMO, outside the city proper so Jon/Dany don't feel like it is a trap, but close enough to the city where Cersei can feel safe and easily bring her guards.

However, we know the actual meeting was organised by Tyrion and Jaime when they met in secret. The details of their conversation isn't gone into in detail but we can speculate that it was organised there and then.

The conversation later on between Jaime and Cersei goes into more detail about the fact that Dany wants to meet with Cersei but that's about it:

Why are you here?
I met with Tyrion.
What did our brother have to say?
Daenerys wants to meet.
To discuss her surrender?
To discuss an armistice.
She's just won a great victory. Why would she want a truce now?
Because an army of dead men is marching on the Seven Kingdoms. Tyrion claims he'll have proof.