What gets counted as "Healing Done" in the League of Legends stats? The scoreboard after a game shows me doing all of 71 healing (some strangely small value), and other non-overt healers doing conspicuous amounts.
The enemy Graves had the summoner spell Heal, which perhaps accounts for his healing, but how did Malzahar or Zilean do that much healing? Does spell vamp count as healing but not life steal (I had Bloodthirster for over half the game and easily healed thousands with it)? Does shielding count (or what part of the shield gets beat on)?
Janna excels in supporting teams.
Howling Gale which launches enemies into the air in a line.
Eye of the Storm shields an ally for a determined amount of damage and gives a damage buff, and it is the only shield capable of protecting turrets.
Moonsoon is a teamfight changer ability as it aoe heals your team and blows away enemies in Janna's ability ratio
Tailwind grants 3% move speed buff! 
Karma can contribute with great magic damage bursts. Additionally, she has a single target movement speed buff. Karma is a special support champ in the sense that she does not posses an ultimate ability per se, but her ult, Mantra, gives her a bonus effect on all of her other 3 spells:
Heavenly Wave + 
Mantra = AOE Heal in a cone in front of Karma
Spirit Bond + 
Soul Shield + 
Kayle makes an ally invulnerable to all damage for 2-3 seconds! Additionally, she heals allies and slows enemies, making her a very nice support pick. Unlike Janna and Karma which build defensive and AP items, Kayle builds Hybrid items like Hextech Gunblade and Guinsoo RageBlade. 
Divine Blessing heals up an ally and grants a movement speed bonus.
Intervention grants invulnerability for 2-3 seconds, which can completely alter the course of a team battle if used well. 
Lux is considered a support/mage. She can tear up teams with her chain of abilities, possesses a 2 unit root, AoE slow, and an amazing Ultimate.
Prismatic Barrier shields an ally for damage. 
Morgana is just like Lux in that she is a support/mage. However, she is considered a better support for the shield she grants:
Black Shield is a very unique shield that blocks magic damage and any negative effects: all snares, silences, blinds, etc. This shield scales with 80% of her total ability power. It is a total carry protector and game changer skill.
Sona is the queen of support in my opinion, closely followed by Soraka. All her abilities except for the ultimate give your teammates a buff and she can not be ignored damage wise. 
Hymn of Valor is an aura granting a buff to damage and ability power. 
Aria of Perseverance is an aura that increases armor and magic resist to allies. On cast, it will heal the most wounded ally nearby. The Powerchord bonus decreases total damage output of the target by 20%
Song of Celerity gives a temporary movement speed buff to your allies, in addition to the aura which grants a fixed movement speed bonus! 
Soraka is one of the oldest support champions, and one of the most powerful ones. She has a single target heal plus armor buff and the ability to grant mana to an ally.
Astral Bleesing heals for a decent amount while granting 20-80.
Infuse grants mana to an ally. No other hero can boast an ability like this.
Wish is a global heal. The epitome of an incredible support heal.
Consecration, her passive, increases magic resistance for nearby allies by 16.
Taric can heal and has an Armor Aura, Damage Aura, and a Stun, which makes him a reliable support character. However, one of his downsides is being a melee character unlike many of the others mentioned.
Imbue, if casted on an ally, heals Taric and your ally. If casted on Taric, he heals himself an additional 40%.
Shatter is an aura that increases allies' armor by 10-30 and increases Taric's by double this amount!
Radiance heals Taric by an amount per second while granting an aura that increases attack damage by 15-45 (and double that on Taric). Make sure to use while taking down towers and in team fights!
Zilean is an amazing support champ with the unique ability to revive champions.
Time Warp is a single target Speed Buff/Debuff
Chronoshift revives a fallen enemy with scaling (200% of AP) restored HP and Mana. That's like fighting a 5.5 vs 5 :)
Heightened Learning, his amazing passive, gives all allies an 8% bonus to experience gained!
Best Answer
I suspect that your 71 HP "Healing" was a result of killing 2-3 neutral creep camp leaders while hurt over the course of the game and receiving the minor heal they now give on death as part of the jungle revamp. The tally shows you as killing 34 Neutral creeps. If you killed about 1/3 of them when you were hurt (so that the heal would have an effect) that comes out to around 71 HP.