Is there a sensible notion of blowing up in any of the available frameworks for derived algebraic geometry? I am happy to remain in the affine setting, where I think the right question to ask is "what does it mean to take powers of an ideal?" in say, a commutative differential graded algebra.

I assume that you're working in the DG category, so I take "ideal" to mean "DG ideal". Let $R$ be a commutative DG algebra and $I$ an ideal of $R$. Then for each $n\geq 1$, I set $I^n$ equal to the intersection of all the DG ideals of $R$ that contain the set $S(n)$ consisting of all elements of the form $a_1\cdots a_n$ such that $a_1,\ldots,a_n\in I$. In other words, $I^n$ is the DG ideal of $R$ generated by $S(n)$. 


I think there was is a problem with characteristics. What is I^n? Is exactly the nth symmetric power of I. In characteristic 0, the symmetric algebra is a direct summand of the tensor algebra, but in positive characteristic I do not think so. 

