Boss Rush: An optional game mode planned since they hit the $2,000,000 goal. Whether or not a Boss Rush occurs during the final levels (a la Mega Man) remains to be seen.
Call Back: The initial funding goal ensures the game having six initial stages before the final one. An additional $300,000 guarantees eight initial stages. As the Kickstarter page's FAQ explains:
In order to set our initial funding goal, we had to ask ourselves what was the cheapest we could make a game that we could still be 100% proud of, that fulfilled all the promises we wanted to make, and that could stand on its own as an evolution of this genre. To find our answer, we actually looked to the past to the amazing original wellspring where this genre first came jumping and blasting its way into this world. Long-time fans will understand, then, when the answer we came up with was "six"; Six stages and main bosses, and then...an additional final stage and challenge...
Character Title: Based on Beck's numerical designation in the Mighty lineup.
Chip Tune: The developers plan to include "retro" music and sound effects as an option.
Chuck Cunningham Syndrome: When it became time to vote on Call's design, the Aero-esque version that was shown beforehand was absent, prompting people to ask where she went. Word of God is that each concept artist was allowed to contribute one design for the vote, and the guy behind that version went with another one.
Co-Op Multiplayer: The $2,750,000 stretch goal is an online co-op version of Challenge Mode where players one and two control Beck and Call through the stage.
Competitive Multiplayer: The $4,000,000 stretch goal, meanwhile, is an online battle race where both players compete to reach the end of the level (the player's opponent is depicted as a ghost image), and an enemy defeated by one player is sent to impede on the other's progress. Both of these goals have been met.
New Game+: Thanks to reaching its third stretch goal, New Game+ (and Turbo Mode) will be added in the game. The Kickstarter page mentions players can carry over their abilities in a harder "second quest".
Power Copying: Taken to new levels compared to the father series where Beck changes his whole body to match his foes powers.
Beck can also take the powers of mooks and use them on-site, similar to the abilities of the Z-Knuckle from Mega Man Zero 4, or Axl's ability to copy an effect or form from any enemy in Mega Man X7 and 8.
The Professor: Three of them, in fact — Dr. White (creator of the Mighty Numbers), Dr. Blackwell (inventor of the Xel technology), and Dr. Sanda (creator of Call).
Punny Name: In the same vein as Mega Man's Rock and Roll, Mighty No. 9 has Beck and Call.
In a meta sense, "Dr. White" puns on his Mega Man counterpart, who is "Dr. Light" in the US and "Dr. Right" in Japan.
Remixed Level: "Challenge Mode" will feature short missions on custom or remixed versions of levels; "Online Beck & Call" mode will be a two-player co-op mode with an added emphasis on teamwork.
Retraux: The $3,800,000 stretch goal (which has been met) is an option to enable 8-bit music and sound effects. Note this does not include retro-style graphics, just the sound.
Robot Girl: Call, created by Dr. Sanda as an assistant for Dr. White.
Shout-Out: Mighty No.9 is a allusion to Cyborg 009, right down to the main protagonist's serial number. One of the boss Mighty Numbers resembles a baby and several of the others appear to uphold National Stereotypes, so time will tell how tight this homage will be.
Another theory about the number, presented by Maximillian, is, Megaman's Japanese name being Rock, or Roku, or 6. 9 is just 6 rotated 180 degrees.
Additionally, the "Mighty" in the title may be a reference to "Mighty Kid", one of two working titles for the original NES game.
The "Mighty" in the title may also be a reference to Tetsuwan Atom (or "Mighty Atom" in English), Astro Boy's original Japanese title, which would make it a combination of the original two classic Japanese robotic Kid Hero manga.
The Kickstarter video is FULL of Mega Man shoutouts. To wit:
The camera scrolling up the side of a building a la Mega Man 2.
A blonde lady and her pink poodle resembling Roll and Rush walk by Inafune (who's wearing all blue) in front of a construction yard.
One of the construction workers is wearing a Met helmet.
From what we've seen of Mighty No. 7 so far, he appears to resemble Haggleman.
The Smurfette Principle: No. 3 is the only female boss. (No. 8 was going to be another one, but the concept artist made a male design that Inafune really liked.)
Spikes of Doom: Concept art for the game shows that these will be an obstacle in Beck's world. He has powers to circumvent their hazards.
Start My Own: A real life example; a bunch of ex-Capcom devs creating a successor to one of Capcom's dormant franchises.