Since I commented so much, let me also try an answer. I find this question somehow difficult, since 'standard' seems so vague to me. [In some sense it is a very long comment, the main message is as in Martin Brandenburg's answer.]

If one asks for a true standard, then, no there is none. But, I think this is a bit much to ask for.

$\mathbb{N}$ is used in the question without explanation so a some sort of standard.

But, is it? Actually, to me there is less of a standard notation for the nonnegative and the positive integers than for the prime numbers (as cryptical expressed in a now deleted comment). Some people use $\mathbb{N}$ to mean the nonnegative integers others to denote the positve integers. (The respective other being denoted by $\mathbb{N}^{\ast}$ and $\mathbb{N}_0$ or something along these lines.) So, basically, one has to define the notation if one wishes to be sure that there is no misunderstanding; or, one uses both nonnegative and positive sufficiently frequently, because each of the pairs then makes clear which is which. [And, indeed, I spent once considerable time to figure out whether in some paper now $\mathbb{N}$ included zero or not; as I needed the result specifically for $0$...at least it did.]

To wit, Henri Cohen in his recent two volume book on number theory refuses to use $\mathbb{N}$ for this reasons and uses appropriate 'decorations' of $\mathbb{Z}$ instead.

And, then there is also the question of $\mathbb{N}$ vs. $\mathbf{N}$ (and rarely, but I just saw it in the paper mentioned by David Speyer, $\mathcal{N}$). Even for the integers there is not a true standard, as the fonts vary in the same way; same for reals, complex, rationals (at least for bf vs bb).

So, why should there be a truer standard for the primes?

And if one ask for something essentially as standard as the above, then I would say yes there is: a capital P in some 'special font' blackboard bold, caligraphic, or boldface. Depending on ones choice regarding the font for, say, the integers with the exception of the concern raised by Seva or the mentioned 'notation-clashes', so that then caligraphic is not at all unusual.

Now, whether or not one should use such a symbol for the primes or not seems to me is a bit of an orthogonal discussion. I'd say it depends on the field, even on the specific paper, and the personal style. In some of my papers I do use it, in some not; depends if it seems useful.

It is true that one is much less likely to use the prime numbers as a 'building block' in some constrcut and thus needs a symbol, while say $\mathbb{Z}^n$, $\mathbb{Z}/n\mathbb{Z}$, and plenty of other things are used often.

But, then why $\mathbb{N}$ (assuming it does not contain $0$)? This could also be avoided
most of the time, or similar arguments could be made.

So, there is not a unique standard, but several standard notations (virtually all some variation on P, namely in blackboardbold, caligraphic, or boldface), which in those fields that need the notation are used frequently; to quote something recent, e.g., Iwaniec&Kowalski have $\mathbb{P}$ as a notation, just like $\mathbb{Z},\mathbb{Q},...$