Ego. Knowing that your name will still be referenced after you're gone means it wasn't all for nothing.

You seem to think there are only two outcomes: completely forgotten, or remembered with profound sadness. Those are the self-imposed limitations of one who can't come to terms with death. Choosing to forget loved ones is a slight to their entire existence. It's time to realign your thinking. Remember those who have passed on with happiness and joy for what they were. Let their memory bring a smile to your face. That's how I remember and how I hope to be remembered. Choose celebration not despair.

Death Is Nothing At All, By Henry Scott Holland

Death is nothing at all.

I have only slipped away to the next room.

I am I and you are you.

Whatever we were to each other,

That, we still are.

Call me by my old familiar name.

Speak to me in the easy way

which you always used.

Put no difference into your tone.

Wear no forced air of solemnity or sorrow.

Laugh as we always laughed

at the little jokes we enjoyed together.

Play, smile, think of me. Pray for me.

Let my name be ever the household word

that it always was.

Let it be spoken without effect.

Without the trace of a shadow on it.

Life means all that it ever meant.

It is the same that it ever was.

There is absolute unbroken continuity.

Why should I be out of mind

because I am out of sight?

I am but waiting for you.

For an interval.

Somewhere. Very near.

Just around the corner.

All is well.

Maybe ask the person who said they wanted to be remembered?

Like anything else, the reasons for something are personal. It's like asking someone why they like chocolate ice cream.

Because that is what emotionally healthy people do when someone they love dies. They remember that person and are aware of the positive impact they had on their lives, and grieve that the positive impacts that this person had on them and the world do not continue. Grieving is hard, but the pain does fade with time, and the positives of the memory eventually overshadow the negatives.

Because the idea of oblivion terrifies some people. What they fail to realize is that in the infinity of time, their names, deeds, and memories will be utterly obliterated and lost long, long before the stars burn out.

Personally I'm looking forward to it.

TL;DR hubris and ego, more or less

You should read the book Denial Of Death by Ernest Becker.

In the book Becker says that the fear of death is always present in human subconsciousness. Resulting in a drive to transcend death at least in a figurative way.

Leave no trace

I'm not sure there is a reason. The instinct to continue existing is so fundamental that it doesn't need any underlying reason, I think. We fear the end of our existence. So if we are remembered after we die, then we still exist in some way.

Most people don't know the difference between themselves and an idea, so saving the idea is saving themselves.

narrator: it wasn't

You can remember people who have passed without being said.

Once you deal with your grief looking back at old photos etc. of loved ones is a really nice activity.

I want to be worth remembering. That's all. If I'm remembered, it means I've done something impactful. Hopefully it's a positive.

Thing is most likely you wouldn’t be in an afterlife or be able to see anything or anyone, you would just cease to exist and you would be doing just as much experiencing and thinking as you did before you were born. So being remembered is a way of extending your existence by living in the minds of those you impacted in your life and knowing you made a difference, that it was worth you being born.

If your name and depictions of you still remain in existence then the concept of you does as well. As long as the concept of you remains that means you can still influence the world with said influence having your "signature", so to say.

Most people like to have the ability to influence the world.

"Tell me I'm a good man."

Alternately:

"Did I make you proud?"

"Every day."

It would be sad to know that people forgot about me although I would never know.

Personally, I don't want to be remembered, except in the context of a few stories. I'm more worried about the ephemeral nature of information in general and the ability of people to manipulate our perception of the past. Imagine if you were well "remembered," but someone just made everything up. That's terrifying to me. It is better to be forgotten than misrepresented, IMO.

Not as an answer to the question, but on the same note: Maybe people could be made forgotten by everyone else upon their death. Like hard-forgetting they ever existed, erasing them, just for the sake of not having to bear the burden of loss.

Relevant: https://m.youtube.com/watch?v=Gm9kOewCpuA

To me it's like wanting any other frivolous thing. Why not? I want a whole pyramid with my name, like back in the day.

I think there are two different things at play there, and both have been mentioned, so all I can add is that it's not one or the other but both. (Well - that and a song)

Partly it's the common human need to feel that we matter - that our lives are in some way significant.

And partly it's the fear of death and the resulting desire to believe that we'll "live on" at least figuratively.

And the song is from Shriekback and is directly on topic - Dust and a Shadow

The vast majority of people being remembered after their deaths were truly bad people, if I am being forgotten I will peobably have done something right