Jax, the Legend, the Champion of 19 Voice Lines
Above is a list of Jax's quotes, a total of 19 quotes. One of these quotes is limited to the Jaximus skin. All of these lines are fairly generic, he has little character to him other than in his taunt, joke, "Surprise! I'm back." and "Let me at 'em". I believe, with a little bit of work, Jax can become the cocky piece of shit we all know he really is.
Jax knows he's good, he references it all the time. He loves to taunt his enemies with how good he is. In my opinion, Jax can become one of the funnier, witty characters with just a bit of work and no changes to his overall character.
I've compiled a small list of taunts and wisecracks Jax could say to individual champions or when certain events happen, such as a double kill or first blood. I will start with these events first.
When Jax participates in or gets First Blood "Did you expect anything different?" (this should be global, or at least to his team+the champion that died)
Jax gets a double or triple kill "It's always nice to remind 'em who the champ is."
Quadra Kill "Those were freebies, I want the challenge."
Pentakill "Damn right. Wooh." or possibly, "Draven was right! It doesn't get much better than this."
Jax gets a Shut Down "Sit down!"
Jax gets a Rampage "Hah hah! Ohoho that's great. You just died to a lamp post! How embarrassing is that?"
An enemy champion goes into stealth near Jax "(As if referring to a small animal) Here boy! Here boy! I'll give you a treat if you just let me bash your head in." or possibly, "Who would've guessed, they ran away!"
Jax survives a fight with less than 10% hp "Guess I'll have to try next time." or possibly, "That was fun as hell! Let's do that again sometime!"
Now to the taunts or individual lines directed at certain champions
"Oh, you promise you won't hurt me? (Chuckles) Damn right you won't!"
"Congrats, you can count to four. Whoop."
After killing Jhin, "I've always liked the number three much more. Fits my style, y'know?"
"Been working on my impression of you, (Screams incoherently about war and death) Do you like it? ...I like it."
"This is my League now, little man. Go back to your show where you kill people who can't fight back!"
"Oh how spooky, you scream at people while they sleep. I make people wake up screaming. Remember that."
"You've worked so hard to regain your 'honor', that little quest stops right here, my little samurai."
"I hope you can run away as fast as you can chase, pup."
"I've found your immovable object and your unstoppable force in human form! It's your boy, Jax!"
"Ooooh. That's a nice hammer. I'll trade this post for your hammer."
"If you focused on hitting me more than hopping around like a rabbit, maybe you could beat me..."
"If you created me, then why am I better than you?"
"Oh shut your yapping and start swinging."
"I can find you a friend, it's called the loser bench."
"Oh no, Jarvan trapped me in his ring of battle! Whatever am I going to doooo?"
"You'll need a little more than a cane in a couple minutes."
"Things are going pretty great for me, thanks for asking."
"You won't like me if I find you again, Wukong."
"How about you go away, adults are talking."
"I love you man, but you really could lose some weight."
"Send Draven my regards."
"You found your worthy opponent, I'm so proud."
These are all the examples I can think of to make Jax more... Jax. If you can think of anything better or come up with one of your own, I'd be happy to see them. I just hope that RIOT can see some of these and implement them at some time.
"Sorry, I don't take students anymore."
"Challenge accepted." (actually this one could be a taunt in general
"Stop hiding and face me like a real man!"
"What is a demigod to the champ?"
"You fight like Demacia."
"Go back to the shadows!"
"Who's that ugly assho- I mean, devlishly handsome paragon."
"Oh, you're here for them? Gimme a sec."
"Can't you be... bigger?"
(in Dragon Form) "Oh THAT'S more like it, haha!"
"Nice fists. Know how to use 'em?"

"You have Scissors. For legs."
"Someone's over-compensating."