The 1987 game of the year is Phantasy Star.
The runner up is Final Fantasy.
The master thread for all of these contests can be found here.

There were enough "Other" votes for a runoff.  The final results of the runoff will be decided tonight, Thursday, Sept 21 at 11:59 PM EST.  Here are the list of candidates.

First are the games that made the cut:

Contra (Arcade)
Konami released this classic into the arcades in 1987.  It was an arcade hit, although many consider the definitive version of the game to be the NES version released a year later.

Maniac Mansion (C64/Apple II)
Maniac Mansion innovated Adventure games in a major way.  This first self-published game from Lucasfilm Games was also the first game to use the SCUMM engine.  Maniac Mansion was designed by Ron Gilbert and Gary Winnick with Tim Schafer getting his first credit as being a play tester for the NES port.

TV Show Intro:  https://www.youtube.com/watch?v=njj-t-WNdag

Final Fantasy (NES)
The first entry in the most successful JRPG series of all time.  Final Fantasy would sell 1.3 million units on the NES and (unlike Dragon Quest) would sell more copies in North America than in Japan.

Mike Tyson’s Punch Out!! (NES)
Minoru Arakawa, president of Nintendo of America, happened to be attending a boxing match of a promising heavyweight boxer named Mike Tyson.  Arakawa was so impressed with Tyson that he immediately made a deal to use his likeness in this game for a mere $50,000.  When the game released in 1987, Mike Tyson was the heavyweight champion of the world.  The fight with him in this game is known to be one of the hardest boss fights in all of gaming.  Mike Tyson’s Punch Out!! went on to sell over 3 million copies and is arguably the most beloved entry in the Punch Out!! series.

Super Mario Bros 2/Doki Doki Panic (NES)
In Japan SMB2 was a difficult Mario game using the same game engine of the first SMB game.  But for the North American release of SMB2, they decided to reskin Doki Doki Panic using the characters Mario, Luigi, Toad and Princess Toadstool.  The North American version of Super Mario Bros 2 went on to sell over 7.4 million copies continuing the Mario Madness of the first game.

Here are the new candidates based on suggestions from the comments:

Mega Man (NES)
First entry in Capcom's iconic Mega Man (or Rock Man) series.  Mega Man was an extremely innovative game for 1987.  You could pick to complete the 6 main stages in any order, although some bosses were much easier to do earlier than others.  When you defeated a boss, then you got his powers.  The game design on the original Mega Man was next level, although most agree that this first game lacked polish.  Also the box art for the North American version was wtf terrible.

Phantasy Star (Master System)
First entry in Sega's classic RPG.  The graphics displayed in Phantasy Star were beyond what you could see on the NES, particularly in the dungeons.

Sid Meier's Pirates! (Home Computers)
The first game to include Sid Meier in the title.  Pirates was doing some amazing, next level stuff in 1987.  It's a huge open world game that wasn't an RPG.  Gameplay involved attacking ships, trading spices, sword fighting, romancing the governor's daughter and other piratey stuff like that.

Ys I: Ancient Ys Vanished (Japanese Home Computers)
Adol Christin begins his adventures with Ys, the longest running series from Nihon Falcom.  Ys I originally released on home computers in Japan in 1987.  It was ported to the Master System and Famicom in 1988, but North America didn't get Ys on the NES.  The most common way to play Ys today is probably bundled together as Ys I and II Chronicles.

Games in the first poll, that didn't make the cut to the runoff:

Double Dragon (Arcade)
Shinobi (Arcade)
Nethack (Home Computers)
Zelda II: The Adventure of Link (NES)

There were a whole lot of other notable games in 1987:

1943 (Arcade)
After Burner (Arcade)
Bubble Bobble 2 (Arcade)
Operation Wolf (Arcade)
R Type (Arcade)
Rastan (Arcade)
Road Blasters (Arcade)
Street Fighter (Arcade)
Dungeon Master (Home Computers)
Sid Meier's Pirates (Home Computers) 
IK+ (Home Computers)
The Last Ninja (Home Computers)
Dragon Quest 2 (NES)
Mega Man (NES)
Metal Gear (NES)
Phantasy Star (Master System)

Another great year. So... Final Fantasy and Metal Gear are in my all time top 5 video game series, but my actual GOTY was from a long gone series called Knightmare.

Konami had an excellent year with releases like Metal Gear, Contra and Super Contra (two mainline contras within a year), Life Force NES and Salamander MSX, Nemesis 2 (the original Gradius 2), The Goonies 2, The Treasure of Usas, and Castlevania 2. But my favorite was the MSX version of Knightmare 2: The Maze of Galious. Here's a good read for the few interested:

https://gamesfromtheblackhole.wordpress.com/2021/05/23/the-maze-of-galious/

La Mulana draws heavy inspiration from it. It even briefly puts you into MoG's world and have you face the first boss, who instead of fighting you, hands you the game's cartridge and throws a peace sign:

The Maze of Galious is the game that ignited my love for exploration, and is the antithesis of the standard modern AAA formula.

Other notable 1987 games that come to mind and understandably aren't mentioned here include Nihon Falcom's Ys1 and Legacy of the Wizard. One of my gaming shames is having never played anything from Falcom.

Bonus: I too was released in 1987 ;D

Ys. Fave game of all time. My fave Yuzo Koshiro OST. Falcom. ARPG. Ys series has enjoyed renewed interest the last decade but for a long time it was mostly unknown outside of Japan. There is even an anime adapted to the first 2 games.

Gotta go with Doki Doki Panic/SMB2.
The mechanics and design were really well done and it had an impressive amount of variety and creativity.

Ouch - that's quite some list of games... I gave NetHack some love, but that's going to be a very geeky choice.  Hey: why shouldn't games be as complicated as brain surgery?

I guess Final Fantasy and Manic Mansion are also very important games. The others are great games, but are just good examples of existing platform-type games, rather than being ground-breaking...

SMB2 was the first Mario platformer I owned (on the GBA mind you) and I do really like the way it looks and that it has multiple characters that play a little differently. That being said I did have kinda mixed feelings on it when I played it a few years back, but it would still have to be my favorite of this bunch.

Of the games there I've played, I think the ones that have aged the best are Contra and SMB2. The original Final Fantasy unfortunately hasn't aged very well at all, and the same holds true for a lot of other games from this time. I think I'll give the nod to Contra this time.

Oh, this was such a good year, so it feels like somewhat of injustice whatever I vote for.
But...Sid Meier's Pirates! is the game I played the most and that I find to be best from that year, along with being quite influential. Open world game with open ended gameplay, leaving you to choose what will you do in your life as a pirate/buccaneer/trader, and what nation(s) will you join or plunder, all the while ingame clock is ticking to your eventual retirement. Really surprised not to see it in main poll.

There were a lot of good games this year, and it was hard to know which ones to put in the top 9.  If I had 2 more spots I would have lncluded Sid Meier's Pirates and The Last Ninja.