Surprisingly, this kind of base system hasn't gotten much attention in the reverse mathematics literature. Note that the system $\mathsf{RCA}_0^*$ proposed by Simpson in §X.4 of *Subsystems of Second Order Arithmetic* is a weakening of $\mathsf{RCA}_0$ in precisely the opposite direction: namely $\mathsf{RCA}_0^*$ plus primitive recursion is equivalent to $\mathsf{RCA}_0$ [D. R. Hirschfeldt, R. A. Shore, *Combinatorial principles weaker than Ramsey's theorem for pairs*, JSL 72 (2007), 171–206; PDF].

I implicitly used a system like you want in *A variant of Mathias forcing that preserves* ACA_{0} [AML 51 (2012), 751–780; arXiv:1110.6559]. This is a function-based system of the form $\mathfrak{N} = (\mathbb{N},\mathcal{N}_1,\mathcal{N}_2,\ldots)$ where $\mathbb{N}$ is the underlying set and each $\mathcal{N}_k$ is a set of functions $\mathbb{N}^k\to\mathbb{N}$ which together form an algebraic clone: each $\mathcal{N}_k$ contains all the constant functions, the projections
$\pi_i(x_1,\dots,x_k) = x_i,$ and if $f \in \mathcal{N}_\ell$ and $g_1,\dots,g_\ell
\in \mathcal{N}_k$ then the superposition $f(g_1(x_1,\dots,x_k),\dots,g_\ell(x_1,\dots,x_k))$ belongs to $\mathcal{N}_{k}$. And this algebraic clone is closed under primitive recursion: there are distinguished $0 \in \mathbb{N}$ (zero) and $\sigma \in \mathcal{N}_{1}$ (successor) such that for any $f \in \mathcal{N}_{k-1}$ and $g \in \mathcal{N}_{k+1}$ there is a *unique* $h \in \mathcal{N}_k$ such that $$h(0,\bar{w}) = f(\bar{w}) \quad\text{and}\quad h(\sigma(x),\bar{w}) = g(h(x,\bar{w}),x,\bar{w})$$ for all $x, \bar{w} \in \mathbb{N}.$ Note that the uniqueness requirement on $h$ is crucial since this is the only form of induction in this system. Strangely, one needs to assume dichotomy $x \dot- y = 0 \lor y \dot- x = 0$ to avoid a few bizarre models. Over this base system, $\Delta^0_1$-comprehension boils down to what I called *uniformization*:

- If $f \in \mathcal{N}_{k+1}$ is such that $\forall \bar{w}\,\exists x\,{f(x,\bar{w}) = 0},$ there is a $g \in \mathcal{N}_k$ such that $\forall\bar{w}\,{f(g(\bar{w}),\bar{w}) = 0}.$

And arithmetic comprehension boils down to what I called *minimization*:

- For every $f \in \mathcal{N}_{k+1}$ there is a $g \in \mathcal{N}_k$ such that $\forall x,\bar{w}\,{f(x,\bar{w}) \geq f(g(\bar{w}),\bar{w})}.$

(The above description is semantic but it is straightforward to formalize the above as a multi-sorted system.)

Similar systems do appear in proof theory (as described in Carl's answer) but only a few are restricted to second-order types. Kohlenbach proposed similar systems in various places. The closest I've seen is what he called $\mathsf{PRA}^2$ in *Things that can and things that cannot be done in* PRA [APAL 102 (2000), 223–245; PDF]. He doesn't actually give a fully detailed description in that paper, but if you chase references you see that it is exactly as the system I described above except that closure under superposition and primitive recursion are ensured by type-2 functionals $\mathbf{S}^k_\ell:\mathcal{N}_k\times\mathcal{N}_\ell^k\to\mathcal{N}_\ell$ and $\mathbf{R}_k:\mathcal{N}_{k-1}\times\mathcal{N}_{k+1}\to\mathcal{N}_{k}$. Uniformization is equivalent to the AC^{0,0}-qf scheme in Kohlenbach's paper. The system assumes quantifier-free induction, which is probably not stronger than the formally weaker form of induction in the system I described above.

missingto get $RCA_0$ is primitive recursion. – François G. Dorais♦ Mar 29 '13 at 17:07