Hi, Tapio

Did you verify the minimum acquisition time, ( 16F88 Datasheet p. 219 ... looks you did !!! ), line impedance between INA out and ADC in, and ref voltage ( or 5v supply ) stability ???

Hope also there's no relay nor "heavy load" driven from the Pic supply ...

Note Thermocouple and INA inputs shielding also have to be really care-treated not to be a spike-antenna.

Need a portion of scheme for further analysis ...

As first trail, try 50 or 100 µS as an acquisition time ... and try to scope the INA output for glitches.
Second trail : a Very-low pass filter between INA and ADC

Third trail : a softw. non-linear-change value rejection !!!

Alain