I'm not familiar with the subject and i don't know if my question make sense.
In Homological mirror symmetry and torus fibrations (http://arxiv.org/abs/math/0011041) a structure of a non-unital $A_{\infty}$-algebra on a $\mathbb{Z}$-graded vector space $B$ is defined by a codifferential $d$ of degree $1$ on the cofree tensor coalgebra $T_{+}(B[1]):=\oplus_{n\geq 1}(B[1])^{\otimes n}$. This differential graded coalgebra is non-unital. The coderivation is uniquely determined by $$ m_{n}\: :\: B^{\otimes n}\to B,\quad n\geq 0. $$ Assume $B=\oplus_{n\geq 0}B^{n}$ is a differential graded algebra, then it carries a non-unital $A_{\infty}$-algebra structure where $m_{1}$ is equal to the differential of $B$, $m_{2}$ is the multiplication and the higher $m_{n}$ are all equal to $0$. Now assume that $B$ has a unit $1\in B^{0}$. It turns out that $B$ is a non-unital $A_{\infty}$-algebra equipped with a strict unit (in the sense of $A_{\infty}$ algebras)!
Is there somewhere a notion of unital $A_{\infty}$-algebra? I guess that the reason for the name non-unital is the fact that $T_{+}(B[1])$ is non-unital as a coalgebra.