For Auld Lang Syne

 is a quest in Fallout: New Vegas.

Quick walkthrough
Talking to Arcade Gannon about his past will eventually reveal his history and the path he wishes to follow in order to aid the wasteland in the upcoming clash between the NCR and the Legion. He will take the Courier on a quest to reunite remnants of the Enclave in the Mojave. Gaining enough trust with Arcade will allow the courier to take this quest and help him resolve issues from his past. Some ways that you can gain trust towards obtaining this quest are:

Visiting the Crashed Vertibird in the Southern Mojave (SE of Nipton). Arcade will say something about not having seen a Vertibird for a while. (Verified)

Talk to Dr. Hildern at Camp McCarran and ask about the OSI.

During Render Unto Caesar (or afterwards, like I did) go to The Fort and once you get there Arcade gets scared. Tell him you just want to talk to Caesar to see what hes up to, then leave.

During the HELIOS quest, direct the power to Freeside

Sometimes walking through the REPCONN site will talk about the site and Pre-War Tech.

Visiting the Nuclear Test Site, East of the Crashed Vertibird will cause him to talk about nuclear dumping in Nevada (unverified if this helps build trust with him).

In relation to ED-E My Love, you should bring ED-E along while doing this quest, as several keyword triggers will be spoken while speaking to the remnants to allow you to progress in that quest.

Detailed walkthrough
Arcade needs to trust you before he will open up about his past. Doing actions that support the Followers of the Apocalypse will help Arcade trust you and open up (Doing the quest That Lucky Old Sun and diverting the power to Freeside will score some trust with Arcade and sometimes conversations can occur if you take him with you to some locations such as the Crashed Vertibird in the southern part of the wastes.) Visiting the REPCONN Headquarters can also cause a conversation to occur if you have 7 or more intelligence. The most beneficial option, however, is to visit the Fort while Arcade is your companion, then remark that you "just want to see what Caesar's up to", as this will instantly earn you enough trust to start the quest, assuming that you have progressed far enough in the main storyline. It seems as long as the first main faction quest, i.e. For the Republic, Wild Card:Side Bets has been completed, this will meet one of the prerequisites for allowing this quest to be taken.

Once you have obtained the quest you will need to visit various locations to recruit 5 old members of the Enclave to aid in the final battle at Hoover Dam. Note that these five members are killable in the world so be careful not to do anything to them should you run into them beforehand or this quest cannot be completed. You can go to the remnants in any order you choose. Your Pip-Boy will direct you to where you need to go. The 5 remnants are: Doctor Henry, Cannibal Johnson, Judah Kreger, Orion Moreno and Daisy Whitman. The last 4 are fairly easy to convince and any dialogue option should bring them to the meeting place, however Henry will take some work if you have not visited Jacobstown and done his questline yet. Guess Who I Saw Today will need to be completed before Henry will agree to go to the Remnant Bunker. Once you have convinced all 5 remnants to head to their secret bunker you will be told to head there and meet them to instruct them as to which side to support in the battle. The passphrases they give you upon recruiting them will be used to open the bunker's main door. You don't need to remember them or unscramble them, once you've sent all 5 there the full sentence is there as a conversation option when you interact with the entry terminal.

Upon entering the bunker head to the briefing room in the back and talk to Judah. You will then have to make a choice, have the Remnants support the NCR or Caesar's Legion in battle. Even if you are working with Mr. House or Yes Man, those questlines still will need them to side with the NCR. Should you ask them to side with the Legion, Cannibal Johnson will leave and there is no way to convince him to stay. You can then talk to Orion for training in Power Armor and will get Cannibal Johnson's power armor since he didn't take it when storming out. You can also ask them to side with the NCR. Orion Moreno will leave because he bears a strong hatred towards them. Leave the briefing room and talk to him in the hanger. You can convince him to stay and fight, with a Speech score of 80, and if you cannot do that, you will have to fight and kill him as he will immediately turn hostile. Regardless of how you handled that situation Judah will tell you to talk to Daisy for Power Armor training and can use the armor you can loot off of Orion if you killed him or armor from Daisy as she says it makes the cockpit too crowded if she's wearing it.

Judah will ask you to head on out and finish up your business while they plan their attack. When the battle at the dam takes place they will be dropped in by the Vertibird and battle whichever side you had them fight against. When you leave the bunker Arcade should approach you and ask what you think he should do. He wonders if he should aid the remnants in battle and follow in his father's footsteps, or stay out and make his own path. If you have him not join the remnants he will give you his father's Enclave Tesla Armor with Helmet for your use. No matter which option you choose, this will complete the quest.

If you are working with Yes Man or Mr. House, then during the battle of the dam you will have to fight the Remnants if you recruit them as there is no option for them to support you.