After the humidity sensor, next to fail was a solar panel, which under the influence of elements started showing signs of wear. A close inspection shows cracks on the plastic cover protective film.
No wonder, it completely lost its ability to convert sunlight into electricity.
Nominally it should output around 6V with no load, and around 100mA under load, but was not able to produce any current when loaded with 40 Ω resistor.
At the same time, the rechargeable battery seems to survive OK, but since I already have it on my desk it will be replaced as well.
Unit is still working, after so many years of operation in wind, snow, haze, and scorching sun.
What failed first is humidity sensor, that was already showing signs of age with it's weird readings as it turned out after all. Here is a picture of the old one:
Datasheet of such sensor HS1101 specifies minimum capacitance to be around 160pF for 0%. No wonder it was giving such erronous readings before. Outdoor unit is equipped with a module that holds also the calibration data (EEPROM with mcu), but fortunately for my unprofessional outdoor humidity measurement, changing just the component gave good enough reading. New one:
After replacement outdoor unit gave a reasonable readings again.