I once asked Jensen this question, when we were at a conference at Oberwolfach.
I told him that I had always assumed that the diamond $\Diamond$ principle was called $\Diamond$ because it expresses that there is an object exhibiting an elaborate degree of internal reflection. After all, if $\langle A_\alpha\mid\alpha<\omega_1\rangle$ is a $\Diamond$-sequence, then it very often happens for $\alpha<\beta$ that $A_\beta\cap\alpha=A_\alpha$, which is an instance of $A_\beta$ reflecting to $A_\alpha$.
But alas, this was not his reason. The actual reason was less interesting, having mainly to do, he said, with him simply needing a new symbol that had not yet been used, and that one was available.
Meanwhile, I believe that we should all adopt my explanation anyway! Henceforth, let it be known that a $\Diamond$ sequence is one exhibiting beautifully captivating internal reflections!