[1] In Japan, this custom is called otoshidama (お年玉), and the envelopes are generally white, not red. I localized it using the similar Chinese custom of red envelopes, which has a more standard English name.
[2] The term 大晦日 (ōmisoka) refers to the traditional Japanese celebration held on the last day of the year, but it can be used roughly interchangeably with the calendar day of New Year's Eve itself.
One day, near the end of the year... Yuuki was walking through the town center of Landsol when he happened upon the members of Twinkle Wish at a street corner.
Hiyori
Ah! it's Yuuki! Long time no see! How've you been? Ah, and how've Peco and the others been? Are you out by yourself today?
Rei
Calm down, Hiyori. He can't answer so many questions at once. Yuuki... I'm glad you seem well.
Yui
Hello, Yuuki. Um... Long time no see. You look well.
Hiyori
Are you out shopping, too? It's almost New Year's, after all.
Yuuki
1. "Shopping?"
Hiyori
Yeah! There aren't a lot of stores in Landsol that are open during the New Year's holiday, so we decided to stock up a little!
Yui
We need to have our own food when the stores are closed, so we bought quite a bit.
Rei
We also needed some decorations for the entrance to our guild house. So, we picked those up while we were at it.
Yui
Rei and Hiyori are going to put the decorations up while I cook up food for New Year's.
Yuuki
1. "Are you preparing for a festival?"
Yui
It's not really a festival... I see, because you've lost your memories, you don't know about the new year...
Rei
So you didn't know... Sorry. I hadn't realized...
Hiyori
... Then, it's like your very first New Year's, Yuuki! Um, it means that this year is about to come to an end!
Yui
1. "What happens when it ends?"
Yui
A new year begins after that. It's not as though the world will end when this year does, so there's no need to worry.
Rei
At the end of the year, we want to sweep away any unhappiness and misfortune from the past year, so we do a lot of cleaning... And on the very last day of the year, we also go to a shrine to pray.
Hiyori
They say that if you welcome the new year with a clean house, good things will happen! We also eat noodles, mochi, and candied chestnuts!
Yui
Ha ha. Hiyori, you're only talking about food.
Rei
It's only natural. I'm also a big fan of your home cooking, Yui. Besides that, it's also a tradition for older people to give children gifts of money in red envelopes.[1]
Yuuki
1. "Red envelopes?"
Yui
The idea is, the money symbolizes your wish for the children to have another year of good health.
Yuuki
1. "I want to give out a red envelope, too."
Hiyori
You mean, to us?
Rei
There should be more of an age difference between him and the person he wants to give a red envelope to. That said, I can think of one person he'd want to give one to.
Yui
Could it be Kokkoro?
Yuuki
1. "She's always taking care of me."
Yui
I see. I'm sure Kokkoro will be happy!
Hiyori
Oh! Then, then! Why don't you come out with us on New Year's Eve?[2]
Rei
On New Year's Eve... I see. It'd definitely be reassuring to have you there. In fact, I'd like you to come along, too.
Yui
If you're there, Yuuki, we'll all be at ease.
Yuuki
1. "What's happening on New Year's Eve?"
Hiyori
Hee hee, that's a secret! But it'll be fun! Here, I'll give you the address! Come here on New Year's Eve! Promise!
As Hiyori gave the address to Yuuki, they made plans to meet on New Year's Eve.