http://www.gamefaqs.com/console/ps2/qna/945498.html?qid=29648
You have to play the game twice, getting the good ending both times, beating all the optional bosses and getting every persona, then at a very precise opportunity in the game you have to go in and out of the velvet room twice. What an absolutely fucking ludicrous task.
 Just sayin.

To my knowledge, the only real requirements for fighting Margaret are it being at least your second playthrough, maxing out her Social Link, and going for the true ending.

No. It needs to be your second playthrough and you need to have gotten the True Ending your first time through. You also need to have maxed out her S. Link. Then, on March 20th after some key things happen, talk to Margaret and she'll wait for you at the end of Heaven.

no friggin dingin way!!!!!!!!!!!!!!

The SMT new game + extras are always ridiculous and anyone with a job and a life should not even attempt to unlock the resident uberboss.

" @Red: You don't need to have gotten the True Ending the first time. "

My playthroughs of the game disagrees, but *shrug*

It's a fun easter egg for big fans of the series and that game in the series. It's not something the player is meant to come across by chance. If you're a big enough fan to go through the game a second time you've probably read how to fight her online anyway.

 
You do have to beat the optional bosses before the end date, but you absolutely do not have to get every persona. That would be...eh. From a certain point of view, you do have to go to the velvet room twice, but that entails going before the gas station thing and going again, the first time you get the note to the challenge (this is how I remember it, anyway). To sum it up: 

Max her social link 
Defeat the optional bosses 
Get True Ending twice 
Talk to Igor in the vevlet room before the gas station incident  
 
Compared to othe SMT games, this isn't very difficult. Stil difficult, but not impossible. 
 
EDIT: I would like to note that maxing her social link the second time through isn't really hard if you registered the personas you used the first time, like I failed to do.

So what happens when you do?  I mean, what is the point?  I'm not down playing the significance, I am just wondering, if I was going to go through this, what is the pay off to make it worth my time?

" So what happens when you do?  I mean, what is the point?  I'm not down playing the significance, I am just wondering, if I was going to go through this, what is the pay off to make it worth my time? "

" @PenguinDust: No real point. SMT games are noted for having extremely time consuming end game bosses, with the optional ones being even harder. In SMT: Digital Devil Saga, you have a chance to fight the HitoShura, aka the main character from SMT 3: Noctune.  He takes about 45 minutes to kill. "

" @fRAWRst: The only hell will be the hell you'll have to pay for shit-staining the one series that still keeps me interesting in gaming. "

" Demifiend sounds stupid as hell to me. Im not here to start a fight, im a fan of SMT too. I hope not everyone here at giant bomb is as tightly wound as you. "

Just curious, but what is the context in which she appears? She just shows up and decides to fight, or is there a purpose in doing so?

You could also fight Elizabeth in Persona 3 FES.

" @Turambar said:

" @Red: You don't need to have gotten the True Ending the first time. "

Yeah. I'm pretty sure you do. "

Wow, so much clashing information here.    
  
So let me get it straight. On the second playthrough, after you go in for the orb you go in again?   
 
... Too bad I'm already at the boss on my 2nd. Gotta save something for the third I guess. Good thing there's a triangle button... Guuuuuuuuh. 

• Obtained True Ending on the first playthrough
• 2nd cycle or later
• Empress social link maxed before finishing Magatsu Inaba dungeon
• The secondary bosses of all the dungeons are defeated before finishing Magatsu Inaba dungeon
• On 3/20,once you followed the True Ending path and faced Izanami, you MUST enter the Velvet Room EDIT: Sorry, looks like I somehow erased the later part of this, but you have to enter the room BEFORE you enter the tv

I posted the videos of the fight in the "Endurance Run the Aftermath" thread, but I guess not everyone has watched the endurance run(crazy i know). The videos are easy enough to find on youtube though. If you don't have some really well-made persona, this fight can take a long time and be pretty annoying. But with the right preparation and patient fusing, you can make it much easier. Which makes some of the more tedious fuses actually worth something at the end of the game.
 
I remember trying to take the cheap way out making a Tri-auto Norn, and it took over 30 minutes of re-rolling it to get repel elec and all 3 autos. Luckily Igor didn't screw me when it finally came up.
 
Some other cool fuses:
 http://persona4.wikidot.com/recipe

@PureRok: 
 ? 
 
That is a source I used heavily and has been cross-referenced many times. It is 100% accurate and verified. 
 
Here is another stating the same thing:  
Click Here    

Elizabeth was a thousand years ahead of her.
 
Main character only
Second playthrough
 
If you used ANY persona with ANY immunity, she would one shot you.
 
Have fun at dinner.

" Elizabeth was a thousand years ahead of her.  Main character only Second playthrough  If you used ANY persona with ANY immunity, she would one shot you.  Have fun at dinner. "

" @Fallen189 said:

" Elizabeth was a thousand years ahead of her.  Main character only Second playthrough  If you used ANY persona with ANY immunity, she would one shot you.  Have fun at dinner. "

Yeah.  Persona 3 was a more difficult game overall, and not just because you couldn't control your party members. "

" @FluxWaveZ: I'd call The Answer difficult.  I played through Persona 3 on Normal Difficulty and had no experience with Hard, which is what The Answer is locked in at.  It requires a lot of grinding to be sure, but even with grinding and fusing, it is significantly more challenging than any portion of The Journey or any portion of Persona 4. "

" @Hailinel said:

" @FluxWaveZ: I'd call The Answer difficult.  I played through Persona 3 on Normal Difficulty and had no experience with Hard, which is what The Answer is locked in at.  It requires a lot of grinding to be sure, but even with grinding and fusing, it is significantly more challenging than any portion of The Journey or any portion of Persona 4. "

Yeah, I had played Persona 3 on Normal difficulty when I had started so I also had no experience with the difficulty set on Hard.  To tell you the truth, most of my memories about The Answer, excluding plot moments, is a little hazy because it was so damn long I listened to podcasts while grinding.  But yeah, now that I think about it maybe most of my frustration was in part due to the difficulty of the whole thing.  Even fusing was hard because there was no Compendium...  Yeah, I guess it was rather hard.  Except for the fights against your teammates: I'm pretty sure all except for one of those was rather too easy for what you were supposed to be facing. "