I could have made this question very brief but instead I've maximally gone the other way and explained a huge amount of background. I don't know whether I put off readers or attract them this way. The question is waay down there.
Let $f$ be a cuspidal modular eigenform of level $\Gamma_0(N)\subseteq SL_2(\mathbf{Z})$ (for example $f$ could be the weight 2 modular form attached to an elliptic curve) and let $p$ be a prime. In the theory of modular forms, one Hecke operator at $p$ is singled out, namely $T_p$, sometimes called $U_p$ if $p$ divides $N$, and defined by the double coset attached to the matrix $\left(\begin{array}{cc}p& 0\\ 0&1\end{array}\right)$. Now $f$ is an eigenform for $T_p$, and $f$ has an eigenvalue for this operator—a Galois-theoretic interpretation of this eigenvalue is that it is (modulo fixing embeddings of $\overline{\mathbf{Q}}$ in $\overline{\mathbf{Q}}_\ell$ and $\mathbf{C}$) the trace of the geometric Frobenius on the inertial invariants of the $\ell$-adic representation attached to $f$, for $\ell\not=p$ a prime.
Now here is a very naive question that I don't know the answer to, and I really should, and I'm sure it's very well-known to people who do this sort of stuff. Say $N=p^rM$ with $M$ prime to $p$. One can approach the theory of Hecke operators entirely locally. Let $K:=U_0(p^r)$ denote the subgroup of $GL_2(\mathbf{Z}_p)$ consisting of matrices whose bottom left hand entry is $0$ mod $p^r$. Now there is an "abstract Hecke algebra" of locally left- and right-$K$-invariant complex-valued functions on $G:=GL_2(\mathbf{Q}_p)$ with compact support. As a complex vector space this algebra has a basis consisting of the characteristic functions $KgK$ as $KgK$ runs through the double cosets of $K$ in $G$. But this space also has an algebra structure, given by convolution.
If $r=0$ then $K$ is maximal compact, and the structure of this Hecke algebra is well-known and easy. Via the Satake isomorphism, the abstract Hecke algebra is isomorphic to $\mathbf{C}[T,S,S^{-1}]$, with $S$ and $T$ independent commuting polynomial variables. The interpretation is that $T$ is the usual Hecke operator $T_p$ attached to the matrix $\left(\begin{array}{cc}p& 0\\ 0&1\end{array}\right)$ and $S$ is the matrix attached to $\left(\begin{array}{cc}p& 0\\ 0&p\end{array}\right)$. One doesn't always see this latter Hecke operator explicitly in elementary developments of the theory because it acts in a very dull way—it acts by scalars on forms of a given weight and level $\Gamma_0(N)$, typically (depending on normalisations) as the scalar $p^{k-2}$ on forms of weight $k$. In particular the "abstract Hecke algebra" doesn't give us any more information than that which classical texts explain, as it's generated by $T_p$, $S_p$ and $S_p^{-1}$.
The next case is $r=1$ and this case I also understand. The abstract Hecke algebra now is non-commutative, "because of oldforms": I don't think the operators attached to $\left(\begin{array}{cc}p& 0\\ 0&1\end{array}\right)$ and $\left(\begin{array}{cc}0& p\\ 1&0\end{array}\right)$ (that is, the operators corresponding to these double coset spaces) commute, but if $f$ has level $Mp$ and is old at $p$ then we should be working at level $M$, and if it's new at $p$ then we get two invariants—the $T_p$ (or $U_p$) eigenvalue, which is classical, and the $w$-eigenvalue, which is the local sign for the functional equation. Again both of these numbers are classical and a lot is known about them. I am pretty sure that the abstract Hecke algebra in this case is generated by these operators $T_p$, $w$, and the uninteresting $S_p$ and $S_p^{-1}$, the latter two still acting by scalars on forms of a given weight. Am I right in thinking that these operators generate the local Hecke algebra? I think so.
The next case is $r=2$ and this I am not 100 percent sure I understand. The classical theory gives us $T_p$, $S_p^{\pm1}$ and $w$. Note that on a newform of level $\Gamma_0(Mp^2)$, $T_p$ is zero in this situation, $S_p$ acts by a scalar, and $w$ is some subtle sign which people have clever ways of working out.
Finally then, the question! Let $K$ be the subgroup of matrices in $GL_2(\mathbf{Z}_p)$ consisting of matrices for which the bottom left hand corner is $0$ mod $p^2$. Let $H$ denote the abstract double coset Hecke algebra of compactly supported $K$-bivariant functions on $GL_2(\mathbf{Q}_p)$.
Is this abstract Hecke algebra generated (as a non-commutative algebra) by the characteristic functions of $KgK$ for $g$ in the set {$\left(\begin{array}{cc}p& 0\\ 0&1\end{array}\right)$, $\left(\begin{array}{cc}0& p^2\\ 1&0\end{array}\right)$, $\left(\begin{array}{cc}p& 0\\ 0&p\end{array}\right)$, $\left(\begin{array}{cc}p^{-1}& 0\\ 0&p^{-1}\end{array}\right)$}?
In the language I've been using in the waffle above: modular forms of level $p^2$ have an action of the Hecke operators $T_p$, $w$, and the invertible $S_p$. Are there any more, lesser known, Hecke operators that we're missing out on?
Best Answer
Just to expand on a comment I made above: I'm not exactly sure what operators generate the Hecke algebra of $\Gamma_0(p^2)$, but the Hecke algebras of the principal congruence subgroups $\Gamma(p^r)$ are easier to handle.
Let's write $K_n$ for the principal congruence subgroup of level $p^n$ in $G = {\rm GL}_2(\mathbb{Z}_p)$.
CLAIM: For any $n > 0$, the Hecke algebra $H(G // K_n)$ is generated by the double cosets $K_n x K_n$ for $x$ in the set
$S = \left\{ \begin{pmatrix} 1 & 0 \\\ 0 & p \end{pmatrix}, \begin{pmatrix} p & 0 \\\ 0 & p \end{pmatrix}, \begin{pmatrix} p^{-1} & 0 \\\ 0 & p^{-1} \end{pmatrix}, \begin{pmatrix} 1 & 1 \\\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 0 & -1 \\\ 1 & 0 \end{pmatrix}, \begin{pmatrix} a & 0 \\\ 0 & 1 \end{pmatrix} \right\}$
where a is your favourite generator of $\mathbb{Z}_p^\times$.
(These correspond, classically, to $T_p$, $S_p$, $S_p^{-1}$, a "twisting operator at p", something close to the Atkin-Lehner $w$, and the diamond operator $\langle a \rangle$.)
Proof: It suffices to show that the subalgebra generated by these operators contains the double coset $[K_n g K_n]$ for any given $g \in G$. Let $X$ be the monoid of elements of the form $\begin{pmatrix} p^r & 0 \\\ 0 & p^s\end{pmatrix}$ with $r \le s \in \mathbb Z$. The Cartan decomposition tells us that any $g \in G$ can be written as $g = k x k'$ for some $k, k' \in K_0$.
We now write $K_n\ g\ K_n = K_n\ k\ x\ k'\ K_n$
$ = K_n\ k\ K_n\ x\ K_n\ k'\ K_n$ (using the normality of $K_n$ in $K_0$)
$ = [K_n\ k\ K_n]\ [K_n\ x\ K_n]\ [K_n\ k'\ K_n]$
The first and last terms are obviously in the subalgebra $H(K_0 // K_n)$ of $H(G // K_n)$, which is isomorphic to the group algebra of the finite group $K_0 / K_n = {\rm GL}_2(\mathbb Z / p^n)$. This is clearly generated by the images of the last three elements of $S$, since these are topological generators of ${\rm GL}_2(\mathbb{Z}_p)$.
Meanwhile, the middle term is in the subalgebra of $H(G // K_n)$ generated by $X$, and it's easy to see that for $x, y \in X$ we have $K_n\ x\ K_n\ y\ K_n = K_n\ xy\ K_n$. Hence this subalgebra is just the monoid algebra of $X$, which is generated by the first three elements of $S$.
Now, as for your original question, the subgroup $U_0(p^2) \subseteq {\rm GL}_2(\mathbb{Z}_p)$ of matrices that are upper triangular modulo $p^2$ contains a conjugate of $K_1$, so its Hecke algebra is isomorphic to a subalgebra of the Hecke algebra of $K_1$. So although I can't give generators for your algebra, I can exhibit it as a subalgebra of something we know generators for.