This was meant to be a comment before it got too long.

One of the ways to define a minimal $A_\infty$ model of a cdga (more generally, an $A_\infty$-algebra) $A$ is Merkulov's recipe (see Merkulov http://arxiv.org/pdf/math/9809172.pdf or Chuang and Lazarev http://arxiv.org/pdf/0802.3507.pdf for a summary). All we need is а decomposition $A=W\oplus K$ into subcomplexes with the differential of $A$ vanishing on $W$, $K$ acyclic, and a contracting homotopy $h:K\to K$ satisfying $h^2=0$. This decomposition is called the Hodge decomposition since it becomes one when $A=\mathcal{E}^*(M)$, the algebra of complex-valued differential forms on a K"ahler manifold $M$, in which case $W$ is the subspace of harmonic forms. Note that $W$ needn't be a subalgebra. Note also that finding such a decomposition is equivalent to splitting $A$ into cocycles and some complement and then splitting off the cohomology inside the cocycles, see Chuang, Lazarev, ibid., p. 5.

So if $A$ is some kind of cochain algebra associated to a complex algebraic variety and we take any mixed Hodge structure on $K$ compatible with $h$ we get higher multiplications $m_n$ on $H^*(A)$ which are maps of mixed Hodge structures of degree $2-n$. In particular the weight of $m_n(x_1,\ldots,x_n)$ will be the sum of the weights of $x_1,\ldots, x_n$ plus $2-n$. Of course there is no reason this $A_\infty$ structure should be compatible with morphisms of varieties. I would guess it is possible to choose the splittings so that it is but this would require some more work.

This was meant to be a comment before it got too long.

One of the ways to define a minimal $A_\infty$ model of a cdga (more generally, an $A_\infty$-algebra) $A$ is Merkulov's recipe (see Merkulov http://arxiv.org/pdf/math/9809172.pdf or Chuang and Lazarev http://arxiv.org/pdf/0802.3507.pdf for a summary). All we need is а decomposition $A=W\oplus K$ into subcomplexes with the differential of $A$ vanishing on $W$ (so $W$ can be identified with $H^*(A)$), $K$ acyclic, and a contracting homotopy $h:K\to K$ satisfying $h^2=0$. This decomposition is called the Hodge decomposition since it becomes one when $A=\mathcal{E}^*(M)$, the algebra of complex-valued differential forms on a K"ahler manifold $M$, in which case $W$ is the subspace of harmonic forms. Note that $W$ needn't be a subalgebra. Note also to find such a decomposition it suffices to find a splitting $A$ into cocycles and some complement and then splitting off the cohomology inside the cocycles, see Chuang, Lazarev, ibid., p. 5.

So if $A$ is some kind of cochain algebra associated to a complex algebraic variety and we take any mixed Hodge structure on $K$ compatible with $h$ and the differential and such that the multpplication $A\otimes A\to A$ is a map of mixed Hodge structures (the latter condition is probably too strong and may be relaxed in some way). Applying Merkulov's formula we get higher multiplications $m_n$ on $H^*(A)$ which are maps of mixed Hodge structures of degree $0$. These maps will be linear combinations of compositions of maps constructed from maps of the form $(0,h)$, multiplications in $A$ and projections to $W$ along $K$. In particular the weight of $m_n(x_1,\ldots,x_n)$ will be the sum of the weights of $x_1,\ldots, x_n$. Of course there is no reason this $A_\infty$ structure should be compatible with morphisms of varieties. I would guess it is possible to choose the splittings so that it is but this would require some more work.