Csdrhrt

In the story, A Game of Thrones, Tyrion Lannister demands a trial by combat after being wrongly accused of trying to kill Bran Stark and conspiring to kill Lord Jon Arryn.

The halls of the Eyrie remain silent for a moment when Tyrion asks for a champion for nobody wants to fight on his behalf until Bronn offers.

Tyrion: Do I have a volunteer?

Laughter and gigges and awkward silence.

Tyrion: No one?

Robert Arryn giggles hoping to see Tyrion defeated and thrown threw the Moon Door.

Tyrion: Anyone?

Lady Lysa: I think we can assume no one is willing to ...

Bronn: I will stand for the dwarf.

Bronn is an amoral and selfish sellsword. When is introduced to the story, he is a freerider more interested in his own comfort and gold than whether others simply live. Much later in the story, he admitted the only reason he saved Jaime Lannister from a dragon's breath was so that Jaime could stay alive until Bronn got his castle from the Lannisters.

He is not moved by claims of honor, so a trial by combat over the rights and honors of hated dwarf of a hated family should hold no appeal to him. I could see him amused to watch Tyrion face combat, but that would be short lived entertainment. I could not see him wishing to participate in that trial.

If he loses, he might die. Defeated but still alive, he might be maimed for life and reviled by the knights of the Vale. If he is maimed, his career as a freeriding sellsword will be cut short.

If he wins, he gains nothing. No gold or promises come to those who participate in a trial by combat. Even if he wins, he might be maimed in the duel. If he wins by killing the favorite knight of Lady Lysa, he will certainly be reviled throughout the Vale.

Both outcomes have very negative outcomes for him. Neither outcome of the trial has a positive outcome.

Why would Bronn offer to fight when either outcome almost certainly has a very high cost for him and no benefit to him?

Please provide answers from the books, but I will accept answers from the TV series.

Well, why do sellswords do anything? For gold.

Since that is a quote, here is Tyrion talking to Bronn as they cross the Vale away from the Eyrie:

“Well, why do sellswords do anything? For gold. You were thinking Lady Catelyn would reward you for your help, perhaps even take you into her service….

“The thing is, you did not know the Starks. Lord Eddard is a proud, honorable, and honest man, and his lady wife is worse. Oh, no doubt she would have found a coin or two for you when this was all over, and pressed it in your hand with a polite word and a look of distaste, but that’s the most you could have hoped for. The Starks look for courage and loyalty and honor in the men they choose to serve them, and if truth be told, you and Chiggen were lowborn scum.”...

My pardons … but you are scum, Bronn, make no mistake. Duty, honor, friendship, what’s that to you? No, don’t trouble yourself, we both know the answer. Still, you’re not stupid. Once we reached the Vale, Lady Stark had no more need of you … but I did, and the one thing the Lannisters have never lacked for is gold. When the moment came to toss the dice, I was counting on your being smart enough to know where your best interest lay. Happily for me, you did.”...

“Scum you may be, but you’re undeniably useful, and with a sword in your hand you’re almost as good as my brother Jaime. What do you want, Bronn? Gold? Land? Women? Keep me alive, and you’ll have it.” Bronn blew gently on the fire, and the flames leapt up higher. “And if you die?” “Why then, I’ll have one mourner whose grief is sincere,” Tyrion said, grinning. “The gold ends when I do.”

-Martin, George R. R.. A Game of Thrones (A Song of Ice and Fire, Book 1)

As he often does, Tyrion had Bronn's number. The script is straight from the books; Bronn stood for Tyrion because he felt it was a good gamble.

In the show, Bronn says "fair enough", in the books he agrees by not disagreeing. "And if you die?" Same difference.

The same scene from the book has a slightly different context

"Sing­er," Tyrion said, turn­ing to Marillion, "when you make a
bal­lad of this, be cer­tain you tell them how Lady Arryn de­nied the
dwarf the right to a cham­pi­on, and sent him forth lame and bruised
and hobbling to face her fin­est knight."

"I deny you noth­ing!" Lysa Arryn said, her voice peeved and shrill
with ir­ri­ta­tion. "Name your cham­pi­on, Imp... if you think you can
find a man to die for you."

"If it is all the same to you, I’d soon­er find one to kill for me."
Tyrion looked over the long hall. No one moved. For a long mo­ment he
won­dered if it had all been a co­los­sal blun­der.

Then there was a stir­ring in the rear of the cham­ber. "I’ll stand
for the dwarf," Bronn called out.

In his second to last quip of this chapter, Tyrion is laying out two things for the audience. He says

When you make a ballad of this

in order to remind everyone of the way news travels in the Seven Kingdoms. It also carries an implication of vengeance against those involved in his execution, by association to "The Rains of Castamere". By this point Bronn has certainly realized that he's not about to be elevated in this company or even receive any substantial reward. Conversely, since he was present at the inn when Tyrion was captured and without the protection of a powerful family Bronn may be worrying he could end up swinging from a tree if he ever ventures into Lannister territory.

Tyrion also says:

to face her finest knight.

This was surely sarcasm on Tyrion's part. Ser Vardis Egen was designated as Lysa's champion moments earlier, and he's about 15 years senior to Bronn (by Catelyn's later estimate). There were dozens of men who volunteered for the duel, many of whom Bronn was sure to avoid fighting, especially Ser Lyn Corbray, established later as a deadly duelist. Bronn is a calculating mercenary, the fact that he knew who he was fighting and felt confident about it surely helped him make his decision.

So, while Bronn's motivation is certainly to obtain gold and riches, like Jolenealaska said, I think there's more to be said for his calculus when he decided to take this risk. He thought it was a safe bet.