I might have a similar issue. I am using AD8237 as a strain gauge amplifier (gain 500). The part seems to have hysteresis at small input voltages. If I pull on the strain gauge, I can see the input voltage change linearly increasing from about -.3mV,-.2mV, -.1mV,0.1mV etc. however the output stays at 1.0mV and stays there as I pull harder and harder until it jumps to like 20 - 30 mV then seems to respond to varying force from there. I am using the Ref pin grounded through a resistor equal to the parallel combination of the two gain setting resistors.
Could this be an offset problem as you detailed for Haox?
It sounds to me like the AD8237 output is saturated for those small or negative input voltages. You can see in the AD8237 datasheet that the linear output range can get to within 20mV from either supply rail under light loading conditions. If the AD8237 is running on a single supply, REF is grounded, and the input is 0mV, then your output would ideally want to go to 0V, which is outside of the linear output voltage range. The circuit wouldn't start to respond until the input voltage was large enough to bring the AD8237 output out of saturation.
The usual solution is to lift the REF pin up by 100mV or so, or whatever it takes so that the AD8237 output is in range for the full expected output range of your sensor (including zero/offset error of the bridge, which can be negative).
You can refer to Figure 72 in the AD8237 datasheet for the simplest way to lift the reference pin, with a resistor divider. It also explains how the gain is affected.
We also have an Instrumentation Amplifier Tool which you can use to see if the AD8237 is in range, and determine how much to lift the REF pin to make sure you are within the linear range: Instrumentation Amplifier Tool - Common Mode vs. Output Voltage (Diamond Plot)
I hope this helps.