I always try to make my games behave this way, too.

The answer can be complicated, or not, depending on what other code might be in your game.

This also depends on which script is printing the text.

Are you using the Followers library? Are you just moving the elf object in a script of your own and just adding that text in said script?

If you are using your own script to print that specific text, and you know it only needs to print "elf" once:

{once:The elf}{notfirst:Adam} enters the room.

That should suffice.

If, though, you have a script which, when triggered, tells the player that the elf's name is Adam, and this script may or may not trigger after the first time the text prints, you could set a boolean attribute on the elfobject to know if the name is known.

Something like this (edited: version 2):

<!--Saved by Quest 5.8.7753.35184-->
<asl version="580">
  <include ref="English.aslx" />
  <include ref="Core.aslx" />
  <game name="elfname">
    <gameid>c03c1a78-33d2-4d9a-a9ba-feda4ccf2922</gameid>
    <version>0.2</version>
    <firstpublished>2021</firstpublished>
  </game>
  <object name="room">
    <inherit name="editor_room" />
    <isroom />
    <object name="player">
      <inherit name="editor_object" />
      <inherit name="editor_player" />
    </object>
    <object name="elf">
      <inherit name="editor_object" />
      <inherit name="namedmale" />
      <alias>elf</alias>
      <usedefaultprefix />
      <speak type="script">
        OutputTextNoBr ("\"Hello,\" says {either elf.alias=\"elf\":the elf:Adam}.")
        if (not this.alias = "Adam") {
          msg (" \"My name is Adam!\".")
          this.alias = "Adam"
          this.usedefaultprefix = false
        }
        else {
          msg ("")
        }
      </speak>
      <look>{either elf.alias="elf":The elf:Adam} is just an ordinary elf.</look>
      <attr name="name_known" type="boolean">false</attr>
    </object>
  </object>
</asl>

If you are using a library that is printing that text, I would add the function to my game's main code and modify it. (If you need help with that, just let us know which library it is.)