What does "on the lease" mean or imply in this context?

Question

In the dictionary, lease means:

A lease is a legal agreement by which the owner of a building, a piece of land, or something such as a car allows someone else to use it for a period of time in return for money.

Therefore, I could not get the logic/implication/meaning of the following sentence, when Stark offered to put her name on the next clean self-sustained tower, but Pepper, his girlfriend, said :

“On the lease".

Does this mean, she's asking for (joking) the ownership of the next tower? Or she asks Stark to rent her the next tower? (But why? Stark does not need to earn the rent money from her.)

What's I am missing here??

Here is the sentence in the context, and the conversation just happened after Stark had lit up Stark Tower for the first time with Arc Reactor powering it:

In the penthouse apartment on top of the new Stark Tower, Pepper hit a switch. The tower began to light up, first from street level, and then hundreds of feet to the pinnacle, illuminating the New York skyline—and the sky itself.

“How does it look?” Pepper asked. He could see her grinning with excitement in the heads-up display inside the helmet of his Iron Man suit.

“Like Christmas, but with more me,” Tony said.

“We’ve got to go wider on the public awareness campaign,” she said, thinking like the CEO she was. The best decision Tony had ever made was handing the reins of Stark Industries over to her. “You need to do some press. I’m in DC tomorrow working on the zoning for the next three buildings. . . .”

“Pepper, you’re killing me,” Tony said as he arced past the Empire State Building. “The moment, remember? Enjoy the moment.”

“Get in here and I will,” she said.

……………………

Pepper stood watching a holographic display of Stark Tower and the Arc Reactor powering it. “Levels are holding steady, I think,” she said.

“Of course they are. I was directly involved, which brings me to my next question. How does it feel to be a genius?”

“Well, I really wouldn’t know, now would I?” Pepper teased him.

“What do you mean? All this came from you.”

“No, all this came from that.” Pepper tapped the miniaturized Arc Reactor in Tony’s chest.

“Give yourself some credit, please. Stark Tower is your baby. Give yourself . . . twelve percent of the credit.”

“Twelve percent.”

Tony had meant it as a joke, but he could see she hadn’t taken it that way. “An argument could be made for fifteen,” he said, trying to get a laugh out of her.

It wasn’t working. “Twelve percent,” she said. “My baby.”

Now he decided to tease her a little more. “Well, I did do all the heavy lifting. Literally, I lifted the heavy things. I’m going to pay for that comment about percentages, in some subtle way later, aren’t I?”

“It’s not going to be that subtle.”

“I’ll tell you what. The next building is going to say Potts on the tower.”

“On the lease,” she corrected him.

The Avengers I

Answer 1

I haven't watched that movie recently so I'm not sure about the scene but I think that the quoted answer provided by @virolino may be not correct.

As you have written, according to the Cambridge Dictionary

lease

a legal agreement in which you pay money in order to use a building, piece of land, vehicle, etc. for a period

On the lease is not an idiom, it means literally "on the lease", on the legal contract signed by Stark with the owner of the building to use it. Stark is not the owner, he's the lesee, the one paying not the one being payed.

Pepper is being sarcastic. She do not believe that Stark is going to put her name in big letters on the façade of the building, because Stark is a "bit" egotist. Instead she thinks that is more probable that he is going to sign the leasing contract using her name, making her responsible of paying money to the landlord. Indirectly she's is calling him a liar.

Notice that they are both joking, playing around. It's a constant between their characters' interaction.

“I’ll tell you what. The next building is going to say Potts on the tower.”

“On the lease,” she corrected him.

Let me rephrase the correction

"No, I'm pretty sure that it's going to say Potts on the lease. So if the Hulk destroys an entire floor, I'm the one who's going to deal with the landlord, the authorities and the insurance company" she corrected him

Answer 2

This answer is the answer:

"

Stark : The next building is gonna say "Potts" on the tower.

"This building says "Stark" on the tower. The next one is going to say "Potts" on the tower"

They're talking about the "next building" after Stark Tower, aka the Tower that has Stark written in big letters outside. So yes, he's saying let's have the next building (with arc reactor energy etc) be Potts Tower, with Potts written in big letters on the actual building.

Pepper : On the lease.

"The next building is going to say "Potts", but only on the lease"

Pepper, otoh, has no interest in having her name in big letters outside a building. She's saying the next tower will indeed have her name on it, but only on paper - she'll own it, but she feels no need to see her name splashed on the outside. Edit: It might also refer to Potts not wanting the next building to be named Potts Tower either, for much the same reasons.

"

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In this context, "on the lease" is synonym with "on paper" (see Dictionary.com):

on paper (Idioms)

• in written or printed form.
• in theory rather than in practice.
• existing only in a preliminary state; in a plan or design:
  • The university building program is still only on paper.

and also (on the same site / page):

on paper

In theory, hypothetically:

• Considering casualties, on paper the Americans won the Vietnam War
• They are a good team on paper but not so in the field.

This metaphoric expression contrasts something written down with concrete reality.