Odin has high AGI and Luck, meaning that he can dodge quite often if you don't have high accuracy. He has about 3 different skills, and they are all deal damage in an area. One of the area attacks is "Zantetsuken", which is a skill that has a chance to inflict instant death. Unsure if there's a pattern as to when he uses it, but it seems random.
The fight may feel like a gamble if you barely have any accuracy, which is why you'll want to prepare yourself with some gears to increase accuracy, or bring characters that have guaranteed hit skills like Holy, Unevadable Pain, or Sharpshoot. After that, be sure to bring VCs to raise area attack resistance like Death Machine, or Netherworldly Crimson Orb, Bloody Moon.
There are plenty of status effects that you can inflict on him, and steal time works as well, so there are plenty of options depending on what status effects your character can use. The important part is to win with no deaths and make 3 or more elemental chains to get the awakening stones. It's easier to do it by focusing on 1 mission at a time.
Recommended Units
Tidus has high damage, has several status effect options, and he can also be the tanker by increasing hate and moving away from the team to avoid team wide damage. But the most important thing is that his limit burst is a very easy way to get 3 elemental chains mission. You can bring just him with any other non matching element and it will still be good. His counter move that can inflict berserk can help make it so Odin stops using area attacks.
Creysse has long range, esper killer from sniper subjob, and a water element unit to help with chains in case you don't want to use Tidus' burst since it's expensive.
Ildyra can provide fast area heals, and deal some water damage as well.
Lightning team is a good option to use since Odin is 30% weak to lightning. Frederika have high accuracy and range.
9S has slash and pierce resist, uses pierce attacks, accuracy increase from passive, ATK + accuracy group buff, and can lower enemy's ATK and MAG, so he's a good unit to bring even if you aren't using a mono lightning team.
King Mont is a pretty good designated tank for the fight. He gets area attack resistances from his master ability, his free VC, and his TMR buff. Death Machine or Bloody Moon if you are not using a mono fire team.
Start by increasing hate, then move away from the team. This may not be 100% reliable since Odin can inflict instant death. It's recommended to bring a unit that can revive, or Yuna since she can use Reraise.