Consider a manifold $M$, and denote by $\Delta _p M$ the set of all submanifolds of dimension $p$ (with or without boundary) of $M$.
Define $G_pM$ to be the free abelian group generated by $\Delta_p M$, and define $\partial : G _p M \rightarrow G_{p-1} M$ to be the linear extension of the boundary "operator" on manifolds: which takes a manifold $M$ and gives back the boundary $\partial M$.
Does this give rise to an homology theory? If so, is it interesting in any way?
There is a homology theory that looks like this called (co)bordism, and it is very interesting. It comes in many flavors depending on what kind of extra structure you ask for on the manifolds.
The basic problem with your proposal is functoriality: the image of a submanifold need not be a submanifold. The correct definition of bordism fixes this by allowing "singular manifolds": that is, we use arbitrary maps from manifolds into $M$, rather than embeddings. Functoriality is then just given by composition.