This hypothesis is equivalent to an instance of $\kappa$, the least point not in $N$, having some degree of extendibility. After all, if $j$ is the inverse of the collapse, then we have an elementary embedding $j:H(\lambda)\to H(\chi)$ with critical point $\kappa$.
A cardinal $\kappa$ is $\eta$-extendible if there is an elementary embedding $j:V_{\kappa+\eta}\to V_\theta$ for some ordinal $\theta$. Although extendibility is usually defined this way in terms of the von Neumann hierarchy, there is a parallel characterization in terms of the hereditary-size hierarchy, and the two hierarchies are interleaved. For example, $V_{\kappa+\eta}$ amounts essentially to $H(\beth_{\kappa+\eta})$, since the von Neumann hierarchy takes the power set each step. There is a good case to be made that the original definition of extendibility should perhaps have referred to the $H(\theta)$ hierarchy instead of the $V_\theta$ hierarchy, since $H(\theta)$ generally satisfies a better theory and has better closure properties. A similar issue affects the strong cardinals, which are also defined by reference to the $V_\theta$-hierarchy.
But meanwhile, if one is in a situation where the least cardinal $\kappa$ not in $N$ is known not to have extendibility properties (which are quite strong in the large cardinal hierarchy), then you know $N$ does not collapse to an $H(\lambda)$. Conversely, when $\kappa$ does have nontrivial extendibility, then you can expect to find such $N$, where $\kappa$ is the least ordinal missing from $N$, and the existence of these elementary submodels for every $\chi$ is equivalent to $\kappa$ being an extendible cardinal.
