One option is to use strace:
strace -o logfile -eopen yourapp
This will log all file-open events, but it will impose a performance penalty that may be significant. It has the advantage of being easy to use however.
Another option is to use LD_PRELOAD. This corresponds to your option #2. The basic idea is to do something like this:
#define _GNU_SOURCE
#include <stdio.h>
#include <dlfcn.h>
int open(const char *fn, int flags) {
    static int (*real_open)(const char *fn, int flags);
    if (!real_open) {
     real_open = dlsym(RTLD_NEXT, "open");
    }
    fprintf(stderr, "opened file '%s'\n", fn);
    return real_open(fn, flags);
}
Then build with:
gcc -fPIC -shared -ldl -o preload-example.so preload-example.c
And run your program with eg:
$ LD_PRELOAD=$PWD/preload-example.so cat /dev/null
opened file '/dev/null'
This has much less overhead.
Note, however, that there are other entry points for opening files - eg, fopen(), openat(), or one of the many legacy compatibility entry points:
00000000000747d0 g    DF .text      000000000000071c  GLIBC_2.2.5 _IO_file_fopen
0000000000068850 g    DF .text      000000000000000a  GLIBC_2.2.5 fopen
000000000006fe60 g    DF .text      00000000000000e2  GLIBC_2.4   open_wmemstream
00000000001209c0  w   DF .text      00000000000000ec  GLIBC_2.2.5 posix_openpt
0000000000069e50 g    DF .text      00000000000003fb  GLIBC_2.2.5 _IO_proc_open
00000000000dcf70 g    DF .text      0000000000000021  GLIBC_2.7   __open64_2
0000000000068a10 g    DF .text      00000000000000f5  GLIBC_2.2.5 fopencookie
000000000006a250 g    DF .text      000000000000009b  GLIBC_2.2.5 popen
00000000000d7b10  w   DF .text      0000000000000080  GLIBC_2.2.5 __open64
0000000000068850 g    DF .text      000000000000000a  GLIBC_2.2.5 _IO_fopen
00000000000d7e70  w   DF .text      0000000000000020  GLIBC_2.7   __openat64_2
00000000000e1ef0 g    DF .text      000000000000005b  GLIBC_2.2.5 openlog
00000000000d7b10  w   DF .text      0000000000000080  GLIBC_2.2.5 open64
0000000000370c10 g    DO .bss       0000000000000008  GLIBC_PRIVATE _dl_open_hook
0000000000031680 g    DF .text      0000000000000240  GLIBC_2.2.5 catopen
000000000006a250 g    DF .text      000000000000009b  GLIBC_2.2.5 _IO_popen
0000000000071af0 g    DF .text      000000000000026a  GLIBC_2.2.5 freopen64
00000000000723a0 g    DF .text      0000000000000183  GLIBC_2.2.5 fmemopen
00000000000a44f0  w   DF .text      0000000000000088  GLIBC_2.4   fdopendir
00000000000d7e70 g    DF .text      0000000000000020  GLIBC_2.7   __openat_2
00000000000a3d00  w   DF .text      0000000000000095  GLIBC_2.2.5 opendir
00000000000dcf40 g    DF .text      0000000000000021  GLIBC_2.7   __open_2
00000000000d7b10  w   DF .text      0000000000000080  GLIBC_2.2.5 __open
0000000000074370 g    DF .text      00000000000000d7  GLIBC_2.2.5 _IO_file_open
0000000000070b40 g    DF .text      00000000000000d2  GLIBC_2.2.5 open_memstream
0000000000070450 g    DF .text      0000000000000272  GLIBC_2.2.5 freopen
00000000000318c0 g    DF .text      00000000000008c4  GLIBC_PRIVATE __open_catalog
00000000000d7b10  w   DF .text      0000000000000080  GLIBC_2.2.5 open
0000000000067e80 g    DF .text      0000000000000332  GLIBC_2.2.5 fdopen
000000000001e9b0 g    DF .text      00000000000003f5  GLIBC_2.2.5 iconv_open
00000000000daca0 g    DF .text      000000000000067b  GLIBC_2.2.5 fts_open
00000000000d7d60  w   DF .text      0000000000000109  GLIBC_2.4   openat
0000000000068850  w   DF .text      000000000000000a  GLIBC_2.2.5 fopen64
00000000000d7d60  w   DF .text      0000000000000109  GLIBC_2.4   openat64
00000000000d6490 g    DF .text      00000000000000b6  GLIBC_2.2.5 posix_spawn_file_actions_addopen
0000000000121b80 g    DF .text      000000000000008a  GLIBC_PRIVATE __libc_dlopen_mode
0000000000067e80 g    DF .text      0000000000000332  GLIBC_2.2.5 _IO_fdopen
You may need to hook all of these for completeness - at the very least, the ones not prefixed with _ should be hooked. In particular, be sure to hook fopen seperately, as the libc-internal call from fopen() to open() is not hooked by a LD_PRELOAD library.
A similar caveat applies to strace - there is the 'openat' syscall as well, and depending on your architecture there may be other legacy syscalls as well. But not as many as with LD_PRELOAD hooks, so if you don't mind the performance hit, it may be an easier option.