Soldato
The crossbow with Steel bolts and the Ice plasmid (Winter Shock or chill iirc) works great on Big Daddies. In fact you become pretty much invincible when you have loads of eve, Winter Chill 2 and a stack of crossbow bolts/mods. I finsihed off an elite bouncer BD with 5 crossbow bolts to the head (modified damage upgrade) after first freezing him
The best thing about big daddies is they don't attack you until you attack them, so make sure you have reloaded all yoru weapons and eve before you start, then when one weapon needs reloading switch to another and get blasting !
Yeah those Elite Bouncers are pretty tough, much stronger than Rosie or the boggo Bouncer... well he done me in more than the normal ones anyway.... and those elite bouncers are fassssst!
As they dont attack until you attack I lay down loads of prox mines in a straight path bewtween me and him. Shoot him with some armour-piecing bullets, as they are not very intelligent they run towards you thru all ya mines... those mines and frag grenades HURT them!
Thats what you can do for later BD's mind you, cos as Aekeron says above, this wont be applicable in ya first BD fight cos you wont have the grenade launcher!