How do I reimplement (or wrap) a syscall function on Linux?

You can use the wrap feature provided by ld. From man ld:

--wrap symbol Use a wrapper function for symbol. Any undefined reference to
symbol will be resolved to __wrap_symbol.

Any undefined reference to __real_symbol will be resolved to symbol.

So you just have to use the prefix __wrap_ for your wrapper function and __real_ when you want to call the real function. A simple example is:

malloc_wrapper.c:

#include <stdio.h>
void *__real_malloc (size_t);

/* This function wraps the real malloc */
void * __wrap_malloc (size_t size)
{
    void *lptr = __real_malloc(size);
    printf("Malloc: %lu bytes @%p\n", size, lptr);
    return lptr;
}

Test application testapp.c:

#include <stdio.h>
#include <stdlib.h>
int main()
{
    free(malloc(1024)); // malloc will resolve to __wrap_malloc
    return 0;
}

Then compile the application:

gcc -c malloc_wrapper.c
gcc -c testapp.c
gcc -Wl,-wrap,malloc testapp.o malloc_wrapper.o -o testapp

The output of the resulting application will be:

$ ./testapp
Malloc: 1024 bytes @0x20d8010