• Create a new Arduino project for ESP8266
  
• Add the "Time: Timekeeping functionality for Arduino" library
  
• Modify the main module of the project like this:
  

  #include <Arduino.h>
  #include <Time.h>
  
  void setup() {
  }
  
  void loop() {
  }
  

  
  
• Run the build. The build fails when compiling the ".../esp8266/time.c" time module:
  

  "C:/Users/stm/.arduinocdt/packages/esp8266/tools/xtensa-lx106-elf-gcc/1.20.0-26-gb404fb9-2/bin/xtensa-lx106-elf-ar" cru  "generic/arduino.ar" "generic/platform/cores/esp8266/StreamString.cpp.o"
  "C:/Users/stm/.arduinocdt/packages/esp8266/tools/xtensa-lx106-elf-gcc/1.20.0-26-gb404fb9-2/bin/xtensa-lx106-elf-gcc" -D__ets__ -DICACHE_FLASH -U__STRICT_ANSI__ "-IC:\Users\stm\.arduinocdt\packages\esp8266\hardware\esp8266\esp8266\2.0.0/tools/sdk/include" -c -Os -g -Wpointer-arith -Wno-implicit-function-declaration -Wl,-EL -fno-inline-functions -nostdlib -mlongcalls -mtext-section-literals -falign-functions=4 -MMD -std=gnu99 -ffunction-sections -fdata-sections -DF_CPU=80000000L -DARDUINO=10607 -DARDUINO_ESP8266_ESP01 -DARDUINO_ARCH_ESP8266  -DESP8266 -I"C:/Users/stm/.arduinocdt/packages/esp8266/hardware/esp8266/esp8266/2.0.0/cores/esp8266" -I"C:/Users/stm/.arduinocdt/packages/esp8266/hardware/esp8266/esp8266/2.0.0/variants/generic" -I"C:/Users/stm/.arduinocdt/libraries/Time/1.5.0" "C:/Users/stm/.arduinocdt/packages/esp8266/hardware/esp8266/esp8266/2.0.0/cores/esp8266/time.c" -o "generic/platform/cores/esp8266/time.c.o"
  In file included from C:\Users\stm\.arduinocdt\packages\esp8266\hardware\esp8266\esp8266\2.0.0/tools/sdk/include/os_type.h:10:0,
                   from C:\Users\stm\.arduinocdt\packages\esp8266\hardware\esp8266\esp8266\2.0.0/tools/sdk/include/sntp.h:4,
                   from C:/Users/stm/.arduinocdt/packages/esp8266/hardware/esp8266/esp8266/2.0.0/cores/esp8266/time.c:20:
  C:\Users\stm\.arduinocdt\packages\esp8266\hardware\esp8266\esp8266\2.0.0/tools/sdk/include/ets_sys.h:148:20: error: unknown type name 'size_t'
   void *pvPortMalloc(size_t xWantedSize) __attribute__((malloc, alloc_size(1)));
                      ^
  C:\Users\stm\.arduinocdt\packages\esp8266\hardware\esp8266\esp8266\2.0.0/tools/sdk/include/ets_sys.h:149:32: error: unknown type name 'size_t'
   void *pvPortRealloc(void* ptr, size_t xWantedSize) __attribute__((alloc_size(2)));
                                  ^
  ...
  

  
  

Well, the compiler is complaining about the missing definition of the type size_t which is generally found in stddef.h

If you think the compiler is trying to drag in the includes for your build machine (vs. the target), go to Project ==> Properties ==> C/C++ General ==> Preprocessor, Include Paths , Macros, etc.. Look at the Providers tab and select and enable the one for the cross compiler (likely the one for the GCC cross compiler). Disable the non-cross compiler builtin providers. You can check them using the Entries tab. The entries on this tab will be searched from top to bottom.

However, I don't think it's possible to have GCC (if that's what you are using) search non-builtins first without totally disabling them.

You didn't include the compiler command from the build log that I can see. However, it does appear to be using the esp8266 headers. Maybe you just need to manually specify stddef.h. If you are letting Eclipse build your makefile you can put additional includes on the command line in tool settings. You may also need to clear your INC_PATH environment variable.

"C:/Users/stm/.arduinocdt/packages/esp8266/tools/xtensa-lx106-elf-gcc/1.20.0-26-gb404fb9-2/bin/xtensa-lx106-elf-gcc" -D__ets__ -DICACHE_FLASH -U__STRICT_ANSI__ "-IC:\Users\stm\.arduinocdt\packages\esp8266\hardware\esp8266\esp8266\2.0.0/tools/sdk/include" -c -Os -g -Wpointer-arith -Wno-implicit-function-declaration -Wl,-EL -fno-inline-functions -nostdlib -mlongcalls -mtext-section-literals -falign-functions=4 -MMD -std=gnu99 -ffunction-sections -fdata-sections -DF_CPU=80000000L -DARDUINO=10607 -DARDUINO_ESP8266_ESP01 -DARDUINO_ARCH_ESP8266  -DESP8266 -I"C:/Users/stm/.arduinocdt/packages/esp8266/hardware/esp8266/esp8266/2.0.0/cores/esp8266" -I"C:/Users/stm/.arduinocdt/packages/esp8266/hardware/esp8266/esp8266/2.0.0/variants/generic" -I"C:/Users/stm/.arduinocdt/libraries/Time/1.5.0" "C:/Users/stm/.arduinocdt/packages/esp8266/hardware/esp8266/esp8266/2.0.0/cores/esp8266/time.c" -o "generic/platform/cores/esp8266/time.c.o"

You can verify that with a couple of gcc commands which will list the full path for includes. Of course you should use the proper gcc version.

This will give make dependencies which list absolute paths of include files but not the full program text. Not sure if they are listed in search order
gcc -M XXX.c ...
If you don't want the system includes (i.e. #include <something.h>), although in this case I think you do, then use
gcc -MM XXX.c ...

also
gcc -H ...

The following gives the program after preprocessing
gcc -E -dI ...

You can go here https://gcc.gnu.org/onlinedocs/gcc/Preprocessor-Options.html#Preprocessor-Options for more information

Have you tried adding -I<path to time.h> before the
-I"C:/Users/stm/.arduinocdt/libraries/Time/1.5.0"?

You at least have a work around if all else fails.

You seem to be saying there is no fix possible other than your work around. Wouldn't this be better posted in a bug report somewhere?
Like here https://bugs.eclipse.org/bugs/enter_bug.cgi?product=CDT
select: cdt-arduino as the component.