Before you get your hopes up, I don't mean physical immortality -- not immortality of the body, or the mind, or the soul. I don't even mean classical "Gilgameshian" immortality -- where people are widely remembered by their cultural works or accomplishments. This latter immortality lacks universality -- only a small fraction of people are remembered, and then only poorly (consider poor Bill Shakespeare and the fact nobody can prove he even wrote his own plays). I'm talking about the related but simpler "remembered immortality" -- being remembered by someone just as a person, not for some works.
How can we achieve this?
Just so that we can keep it clear, let's introduce the idea of a "personal celebrity". A personal celebrity for you is somebody who is "famous" to you -- your relationship to them is just like your relationship to a regular famous person. You know some stuff about them.
We will achieve universal immortality by making sure that every human ever is the personal celebrity to somebody at every moment in time, forever. At every moment, somebody, somewhere, will remember every single one of us: know our name, what we did, what we liked, a few things about us.
To simplify things, we'll ignore everybody who lived before now. (It's possible to rework this to account for them, but it's a lot of bother, and really, its their own fault for not thinking of this back then!)
Suppose you have a list of personal celebrities, and suppose it's unique -- nobody else in the world has that list of personal celebrities. You get married (or whatever), and you and your spouse have two kids. Who should your kids get as their personal celebrities? Well, once you're dead, nobody will remember anyone from your list. So you need to pass your list on to your kids. But so does your spouse! So you each give your list to one of your kids to keep the memories alive. That's not good enough, though, because once you're dead, who will have you as their personal celebrity? Nobody. So you need to pass your list on to your kid, but with one extra name.
That won't work. After enough generations have passed, the list of names will get too long. Remember, you actually need more than just the list. You have to learn all about every name on your list, and remember them well. So we need to keep the list short.
Fortunately, there's an easy solution: you just need to have more than two kids.
Also, to simplify the math, let's assume you're on your own list (you might as well be, you know everything about yourself anyway). We also want to add some redundancy--we want everyone to be on more than one list. So we'll always have each person including him or herself on their own list, and all of their siblings.
Now, suppose your personal celebrity list (including your siblings) has eight names on it. You get married, and you and your domestic partner have four kids. Between you and your partner, you have 16 personal celebrities to pass on to your kids. You divide them up evenly; each child gets 4 personal celebrities from your lists, and each gets 4 personal celebrities by including their siblings and themselves. Each child ends up with 8 personal celebrities, exactly the number we started with.
Every generation after that, someone out there will remember every personal celebrity. (In fact, four people will.)
Now, I know what you're thinking, everyone always having four kids is a lot.
To you I say: but think of the benefits!
But even so, I'm willing to grant that the above might be a little excessive. Fortunately, we can make the same thing work differently. If we all keep a personal celebrity list with nine celebrities, then each couple only needs to have three kids. The couple divides 18 personal celebrities into thirds, giving each child 6 inherited celebrities, plus the 3 siblings-and-themselves makes 9 again.
That's all there is to it! The secret of immortality unlocked!
All we need is for every human ever to have exactly three kids.
Bonus: Don't trust the frailty of human memory? We can all write a short (4-page?) biography and deliver it to our siblings. Then everyone is charged with recopying the biography of all of their inherited personal celebrities. Recopying avoids problems with text aging, and allows translation into new languages as they arise.