LT1677CN8 alternative?
The LT1677 is a low-noise, rail-rail, precision* operational amplifier.

Looking at the circuit, I think you need the low-noise part first, and maybe the rail-rail part.  Based on the circuit you show, bias is taken care of, and based on the link that shows the output coming from a coil, the input impedance to the amp is small.  So the most important parameter is the voltage noise.

So -- anything that is rail-rail, works at 5V, and has voltage noise roughly that low.  If you're in a pinch, you can probably use any rail-rail amp, or even an LM358 if you adjust for the pinout and set the center voltage a bit lower by adjusting R4 and R5 (I'd aim for 1.8V with a 5V supply).

* Ever notice how they're all "precision"?