Blue Mage would use a weapon like a scimitar while also being able to cast through the weapon. It could be like a melee caster. They could learn skills through job quests, this way SE could bring more iconic monsters to learn skills from. The questline would progressively lead to more powerful skills from stronger creatures.

As for skills I was thinking of a sort of stance set up. Five or six stances that would be based on a monster you're learning skills from. So the idea would be you could learn one to say for skills from one type of monster. The skills could be incorporated through the weapon to create a magic sword effect.The Blue Mage's skillset would have a large range of uses from DPS to saboteur. We don't really have a class that can severely debilitate enemies. Blue Mage could have the potential to learn the monster skills we've come to hate. Like a coeurl blaster that adds multiple status effects, or a petrifying gaze that inflicts stone for one to two seconds to enemies infront of you.

Their questline could be a bounty hunter type quest. You would essentially hunt and learn skills from a variety of monsters. Each stance would have its own quest line for the specific monster, leading up to stronger enemies in that stance's questline. The final quests for 50-60 could have mini versions of boss character skills.

As for attire they should have light gear with capes or cloaks. Perhaps a Thavarian theme for blue mage? Thavarian gear reminds me of Blue Mage from FFTA. I like the cape/turban combo with the feather for Blue Mage.

As for strengths and weaknesses I think Blue Mage should have more defense than magic dps and ranged dps, but less than the melee jobs. Blue Mage would have skills that would alleviate this though.

Sorry if its a little jumbled. I just typed out whatever came to me.