I have no idea at the moment where to find a reference for the specific result you seek. However, it can be deduced from the following fact: topological manifolds (paracompact and Hausdorff) are absolute neighbourhood retracts, and thus have the homotopy type of CW-complexes. This is briefly stated in corollary 1 of Milnor's article [*On spaces having the homotopy type of a CW-complex*](http://www.ams.org/journals/tran/1959-090-02/S0002-9947-1959-0100267-4/) (published in Trans. Amer. Math. Soc. 90, 1959, pages 272-280). An extensive discussion of these matters is given in the thesis [*A topological manifold is homotopy equivalent to some CW-complex*](http://image.diku.dk/aasa/oldpage/aasa.pdf) by Aasa Feragen. For the specific case of compact manifolds, an elementary proof is given in the appendix of Hatcher's book *Algebraic topology* (see corollary A.12 there). <!-- Thanks to the now deleted answer of Francesco Polizzi for reminding me of this source. -->

<!--Though interesting, this is unnecessary...
2. The inclusion of the boundary of a manifold is a Hurewicz cofibration, as follows from the collaring theorem, proved by Morton Brown in [*Locally Flat Imbeddings of Topological Manifolds*](http://www.jstor.org/stable/1970177) (published in the Annals of Mathematics, Second Series, Vol. 75, No. 2, 1962, pages 331-341). This result was later given a simpler proof by Robert Connelly in [*A new proof of Brown's collaring theorem*](http://www.ams.org/journals/proc/1971-027-01/S0002-9939-1971-0267588-7/), published in Proc. Amer. Math. Soc. 27, 1971, pages 180-182. The collaring theorem implies the existence of a collar of the boundary, which shows that the pair $(M,\partial M)$ is a NDR-pair, and thus $\partial M \to M$ is a cofibration.
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We can now prove the result you state. Let $Y$ be a CW-complex admitting a homotopy equivalence $f : Y \to \partial M$ to the topological manifold $\partial M$. Denote by $j : \partial M \to M$ the inclusion of the boundary of $M$, and factor the composition
$$ Y \overset{f}{\longrightarrow} \partial M \overset{j}{\longrightarrow} M $$
as the inclusion of a sub-CW-complex $i : Y \to X$ followed by a weak equivalence $h : X \to M$. The weak equivalence $h$ is a homotopy equivalence since both its domain and its target have the homotopy type of CW-complexes.

Finally, let me briefly justify why we can take $i : Y \to X$ to be the inclusion of a subcomplex of a CW-complex. The usual construction of the factorization above produces $(X,Y)$ as a *relative* CW-complex, i.e. $X$ is obtained from $Y$ by adding cells in increasing order of their dimension. Now observe that this construction can be modified to inductively build $X$ as a CW-complex itself: at each stage of the construction of $X$, we can replace the attaching maps of the cells with homotopic cellular maps using the [cellular approximation theorem](https://en.wikipedia.org/wiki/Cellular_approximation_theorem).