Due to Cerf, there's exist a certain homotopy between two Morse functions. It is said that this homotopy is "generic". What is a precise definition of the property to be "generic" in this case?
Intuitively, for the homotopy to be generic has the same meaning as for the Morse function itself to be generic: the set of points where it fails to be a submersion is as simple as possible. Practically, this means that for all but a finite number of times (values of the $t$ parameter) the function $H(x,t)$ is a Morse function of the variable $x$ and the Morse singularities trace out smooth curves transverse to the $x$ direction; and for each of those exceptional times $t$ the function $H(x,t)$ is Morse at all values of $x$ except for a single value $x$ (I'm assuming compactness here) where it undergoes one of a certain class of very special singularities obtained by collapsing two Morse singularities, called "birth-death" singularities. Just as with Morse singularities themselves, which are described locally by specific functions of $x$ in some coordinate system, birth-death singularities are described locally by specific functions of $x,t$ in some coordinate system. The function $H(x,t) = x^2 - t$ at $t=0$ is an example of a birth-death singularity in one dimension.
"Generic" usually refers to open and dense.
Assume $M$ is a closed smooth manifold. Let $$C^\infty(M)$$ be the space of all smooth real valued functions. Topologize this with respect to the Whitney $C^\infty$ topology (see: http://ncatlab.org/nlab/show/C-infinity+topology)
Let $$H(M) \subset C^\infty(M)$$ be the subspace of functions whose singularities are either Morse or birth-death (in particular, $H(M)$ contains the space of Morse functions). Then $H(M)$ is a generic subspace (it's open and dense). It is also path connected. Even more is true: the complement $\cal D$ of $H(M)$ inside $C^\infty(M)$ has codimension two, in the sense that any smooth map $D^1 \to C^\infty(M)$ whose endpoints are in $H(M)$ can be infinitesimally perturbed, relative to its boundary $S^0$, to a smooth map that has image in $H(M)$. Here smooth means that the adjoint map $M\times D^1\to \Bbb R$ is a smooth map.
This is discussed in beautiful detail in the book:
In any case, note that $C^\infty(M)$ is a contractible space (in fact it's affine). This means that if we are given a smooth map $S^0 \to H(M)$, then it admits an extension to a map $D^1 \to C^\infty(M)$. By the above discussion, this map can be perturbed slightly, relative to $S^0$, so that its image lies in $H(M)$.