What is the name of this circuit?
Yes - it has a name. In control theory this circuit is known as a PD-T1 unit. It has a proportional-derivative behaviour with a certain delay term T1. In filter terms, it works like a first-order high-pass with a superimposed constant gain.

The transfer function is \$H(s)= 1 + sR1 \cdot \dfrac{C}{1+sR2C}\$

This device is used to enhance the phase (for stabilizing purposes) in a certain frequency range. Please note that application as a PD-T1 element requires \$R1&gt;R2\$.

More than that, the shown circuit is used as a simple non-inverting amplifier (gain: \$1+R1/R2\$) for single-supply operation. For this purpose, the non-inv. input is dc biased with 50% of the supply voltage - with the consequence that the input signal must be coupled via an input capacitor. Because the dc gain remains unity, the bias voltage is transferred to the output also with the gain of "1".

BODE diagram: The magnitude starts at unity and begins to rise at \$wz=\dfrac{1}{(R1+R2)C}\$, then it stops rising at \$wp=1/R2C\$ at a gain value of \$1+(R1/R2)\$. The rising of the gain is connected with a corresponding phase enhancement.

Because of the mentioned phase enhancement properties the PD-T1 block is also known as a "lead controller".