Unity Gain Op-amp - 0 volts input giving 1.x volts output. Why?
You are lucky to be seeing any behaviour similar to an op-amp at all at that rail voltage. It really prefers to run at +/- 15v, this is an old-skool op-amp.

If you look at the OP27 data sheet, you will see the 'input voltage range' specification is typically +/- 12.3v with a +/- 15v supply, which is 2.7v away from either rail. So the typical op27 doesn't even have a valid input range when supplied with less than 5.4v. The guaranteed specification is +/- 11v, so no valid input range for less than an 8v supply. If you look at figure 12 on page 9, they don't bother to plot a supply voltage of less than 7v.

The output voltage swing is also limited, typically +/- 13.5v with a light load, so 1.5v away from either rail, which is perfectly adequate when you have the headroom of 15v supplies.

Have a look at the functional block diagram on the first page of the data sheet to see the reasons that the input and output range get nowhere near to the rails.

If you want to have outputs or inputs close to a rail, you need to choose a rail to rail (R2R) opamp. LM324 is not true R2R, but it will run on 5v and get you to the -ve rail, however it still won't get within 2v of the positive rail.