Low cost op-amp with Common-mode input voltage range of 1.125V
If you need to only use a single supply as shown but still must be able to handle inputs which are close to ground, I would suggest looking for a rail-to-rail op amp, which are specifically designed for their inputs and/or output ranges to swing very close to both supply voltages. There are R-R input op amps, R-R output op amps, and there are both (typically referred to as RRIO op amps).

However, all this to say that most modern op amps, even non R-R input ones, should be able to handle the common mode input you describe, which is over a volt above ground. It sounds like you might want a precision DC op amp for your purposes, and there are lots of good options. Chopper/autozero amps effectively get rid of their intrinsically-large low-frequency noise &amp; drift through internal switching methods which essentially shift the 1/f noise content to the switching frequency, making them excellent for precision DC measurements. 

I’d suggest a part like the LTC2057HV, which is a great low-noise, precision DC op amp with a wide supply range and a CM input range that actually can handle even 100mV below its negative supply! The data sheet is here: https://www.analog.com/media/en/technical-documentation/data-sheets/2057f.pdf