Dtjytyk

In the movie The Piano, Ada is a mute, who likes to play the piano. When she's married off to Stewart, she insist on taking the piano with her.

Baines, Stewart's friend, buys the Piano from him against her will. When Baines offers up the Piano to Ada in exchange for sexual favors, she's willing to indulge his every request.

Why was the Piano so important to Ada that she was willing to do anything to get it back?

Jane Campion, from The Independent (article by Quentin Curtis, October 1993):

She is sympathetically out of step with the rest of the world: too extreme, too purist, too wild - particularly in Victorian times - to be understood. She has too high motives, and the world will always disappoint her. The piano enables her to have a voice, becoming almost indistinguishable from herself. Their fates are linked. And because she doesn't speak, her intimacy with people becomes more instinctive and physical. Which takes us into the area that the film is exploring about eroticism and fetishism.

And on the relationship between Baines and Ada:

'He never forces her to do anything,' Campion says. 'He may start by pushing her, but he finds it important for her to come to him willingly. He wants to be loved: which is what separates him from a psychopath, who wouldn't believe that anybody could love him.'

Ada, a mute widow in Victorian times, is sold into a marriage in a foreign country and can only express herself through her piano. When her piano is taken away from her - against her will - she wants to get it back. She has nothing, except for her piano and her daughter. For sure, it seems like Baines exploits her, but as Campion notes, he loves her and she is actually willing.

On why that piano specifically (source is the 4thd draft of the script).

1) It's a special piano

Ada developed a special relationship to her piano. That's not unusual, as Christian Jarrett writes in the psychologist:

As our lives unfold, our things embody our sense of self-hood and identity still further, becoming external receptacles for our memories, relationships and travels.

Probably even more so for Ada, since she is so dependent on it. She needs it to express herself, so it's basically her voice. And a longstanding "relationship" (her husband died, her daughter is only 10 years old) - in scene 35, Stewart says he has it in a letter that she has been playing piano since she was 5 or 6.

From Scene 9:

Her daughter is on all fours evidently being sick. But ADA's attention
is diverted to the seamen who are staggering through the waves with a
huge piano shaped box. They put it down as soon as they get to 5hore
but ADA makes gestures that they must immediately bring it to higher
safer ground. The piano placed to her satisfaction she hovers near it,
one hand in constant touch of it while her daughter grips her free
hand.

Or in scene 36 (Stewart tells her about the deal):

Ada "finger signs, her face furious." and then Flora translates

She says it's her piano, and she won't have him touch it. He's an oaf, he can't read, he's ignorant.

and then

ADA's breathing becomes heavy with anger, she writes Curiously on her
pad.

NO! NO! THE PIANO IS MINE! IT'S MINE!

It's also a special piano in itself, see scene 43 where the blind man, who should tune it, tells Baines:

Ah, a Broadbent. A fine instrument. I've not come across one here, or in the Islands where I have tuned some 200. Yes, they like their pianos there.

2) The only piano she could get

Ada is mute, in a foreign country where she knows no one (except for her daughter), including her husband. Even though her piano would be the easiest to get - it's already hers, you only need to pay for transport, the transport crew is already there - but still Stewart refuses (read scene 16). When Ada plays the kitchen table like it was a piano, Stewart notes (scene 34):

I knew she was mute, but now I'm thinking it's more than that. I'm wondering if she's not brain affected.

It's clear that he doesn't care (and trades it to Baines without consulting her, scene 35). How would she have gotten another piano? Would Stewart have given her money, is there a town in 1850s New Zealand where she, together with her 10-year-old translator, could buy one and then transport it to their house? If Stewart had been willing, all could have been much easier. But her husband does not care.

3) The piano easiest to get

This is from a different angle - the piano is still the closest and easiest one to get. She knows who has it (Baines) and is even forced to stay with him to teach him (Stewart at the end of scene 36: "You will teach him. I shall see to that!"). And Baines loves her (unlike her husband!). It looks like exploitation at first, but in a way Ada turns the tables and uses it to her advantage and to exert power over him. She can get back a piano, her piano, by herself.

Regarding the way she gets it back, the sexual favors, allow me a short digression to illustrate the point. When talking about The Wolf of Wallstreet, Margot Robbie said (Robbie Collin, The Daily Telegraph, 2014 from the Business Insider):

The whole point of Naomi is that her body is her only form of currency in this world. So when Marty was trying to help me out, and said in the scene where she seduces Jordan perhaps I could have a robe on, I said she wouldn't. She has to be naked. She's laying her cards on the table.

That's even more the case for Ada - that's what she can offer a man who is in love with her: Intimacy.