By invoking your Sixth Amendment right, if you are charged with a crime and the prosecutor wants to use your invocation of that right against you, you will probably be able to keep that information away from the jury under the law, because the federal courts (at least so far) generally agree that you cannot tell the jury that the defendant has asserted the Sixth Amendment right to a lawyer, or to use that as evidence against the defendant.2 And even if you cannot keep it out of the evidence at trial and the jury is allowed to learn what you said to the agent, it will sound far less suspicious if you merely told the officer that you wanted a lawyer present before you agreed to be interviewed. That makes it sound, after all, like you were willing to answer their questions.
James Duane in You Have the Right to Remain Innocent