The Revised Bladesinger v1.3
Bladesingers are cool! They weave magic into their flashing blade and enhance their physical prowess, striking with swiftness and surity that few can match. In theory, anyway. The bladesinger subclass that exists is, in my view, both overpowered in some ways, doesn't match up to its promises in others, and generally doesn't feel like it fits with 5e's design principles. My chief problem with the class was that its core feature broke bounded accuracy, giving a flat bonus of your Int to both your AC and Concentration checks. Additionally, both Training in War and Song (granting light armor) and Song of Defense were just as useful to a spellcasting wizard as to a gish.
The New Version
So, what's my new Bladesinger? Well, at its core, it's based on the idea of the Song of Defense feature - spending spell slots as a resource in order to fuel your abilities. This makes a great deal of sense to me as a way to transform a full caster into a martial class. But I also wanted to leave the bonuses of Bladesong active all the time, so that if you run out of spell slots you don't have to run off the frontline. I also wanted to make sure that it encouraged you to be a gish, rather than just a caster.
Arcane Tradition
Bladesinging
Wizards have the following Arcane Tradition option, in addition to those in the Player's Handbook.
Training in War and Song
When you adopt this tradition at 2nd level, you gain proficiency with one melee weapon of your choice that does not have the heavy property.
You also gain proficiency in the Performance skill if you don't already have it.
You learn either the Booming Blade cantrip or the Greenflame Blade cantrip, and the spell you choose does not count against your cantrips known.
Your hit point maximum increases by 2, and increases by 1 every time you gain a level in this class.
Bladesong
Starting at 2nd level, while you are wearing no armor and not wielding a shield, your AC equals 10 + your Dexterity modifier + your Intelligence modifier.
Additionally, you can take the Dash or Disengage actions as a bonus action.
When you hit with a melee weapon attack, you can spend your reaction to add your intelligence modifier to the damage of the attack.
Song of Defense
Starting at 6th level, you can direct your magic to absorb damage. When you take damage from a weapon attack, you can use your reaction to spend one spell slot and reduce the damage to you by an amount equal to 2d6 per level of the spell slot spent + your Intelligence modifier.
Additionally, when you fail a Dexterity or Constitution saving throw, you can use your reaction to spend a spell slot and reroll the saving throw with a bonus equal to the level of the spell slot spent.
Song of Attack
Starting at 10th level, when you cast a spell that requires a saving throw, you can channel it into your blade. Instead of choosing the spell's target normally, make a melee attack with a weapon against a single target. If you hit, they suffer the attack's normal effects and are targetted by the spell cast, and must make their save at disadvantage. The spell takes effect after the damage of the attack is dealt.
Song of Victory
Starting at 14th level, when you reduce an enemy to 0 health with a melee weapon attack, you may spend a spell slot of 4th level or below to regain an expended spell slot one level higher than the slot spent. You may use this feature a number of times equal to your Intelligence modifier (a minimum of once). You regain any expended uses when you finish a long rest
Final Notes and Explanations
Training in War and Song hasn't changed much. Mainly, it no longer gives proficiency in armor, since you'll be using the unarmored defense granted by Bladesong instead, and it allows a wider variety of weapons. It also gives +1 HP per level to help your wizard be slightly beefier. I have also added that it teaches a SCAGtrip to help encourage gishing, as many of the low-level features are good for any wizard, not just a gish.
Bladesong, on the other hand, is very different. Instead of a flat AC bonus, you receive an Intelligence-based Unarmored Defense. This fits into 5e's bounded accuracy better. Increased mobility is represented by the ability to Dash or Disengage as a bonus action, which seems to fit into the class's paradigm. I considered giving the Dodge action, but upon getting feeback decided that that a) that would stepping on the toes of the Monk too much and b) would be too strong for a class which rarely has use for its bonus action. It no longer requires a particular category of weapons in order to allow for concepts like a githyanki who uses a greatsword, but you'll have to pick up proficiency in the heavier weapons elsewhere. Investing a feat or a level in another class in order to gain those proficiencies seems like a fair trade to me. You can add your Int to damage once a round now - this is a nice boost early on, although it will be competing later with Song of Defense as well as Counterspell.
Song of Defense is shifted to 6th level and now comes with a bonus to saving throws. Additionally, its damage reduction feature works only against weapon attacks. This is to make sure that the class is primarily geared towards gish characters, rather than just being a general bonus to the survivability of a wizard.
Song of Attack is inspired by the duskblade from 3.5. It lets you force saves with disadvantage, but in return you can only choose a single target, even for a spell like Fireball. Additionally, if the attack misses, the spell slot is expended and the target doesn't even suffer half effect. It should be noted that Song of Attack does allow you to channel cantrips which require saving throws as well as leveled spells. This is intentional, helping you to keep your melee damage up when SCAGtrips like Booming Blade begin to fall off.
Song of Victory is a way to recover some of the mid-level spell slots you spend. It has a limited number of uses and is limited to nothing higher than 5th level for what should be obvious reasons. Note that it can't give you more than four extra 5th level slots (because you don't get more than three 4th level slots, and one of your at-most five uses would need to be used to turn a 3rd level slot into a 4th level slot before you can turn it into a 5th level slot.