As all music copyright holders will tell you, adding music you like (but do not own) to family video clips is copyright infringement. As such, you should remove the audio track entirely to avoid getting into a lawsuit… or worse, getting your video removed from Youtube :)

The command below is will list all streams that exist in your video file.

$ ffmpeg -i yourfile.mp4

ffmpeg version N-60592-gfd982f2 Copyright (c) 2000-2014 the FFmpeg developers
  built on Feb 13 2014 22:05:50 with gcc 4.8.2 (GCC)
  configuration: --enable-gpl --enable-version3 --disable-w32threads --enable-avisynth --enable-bzlib --enable-fontconfig --enable-frei0r --enable-gnutls --enable-iconv --enable-libass --enable-libbluray --enable-libcaca --enable-libfreetype --enable-libgsm --enable-libilbc --enable-libmodplug --enable-libmp3lame --enable-libopencore-amrnb --enable-libopencore-amrwb --enable-libopenjpeg --enable-libopus --enable-librtmp --enable-libschroedinger --enable-libsoxr --enable-libspeex --enable-libtheora --enable-libtwolame --enable-libvidstab --enable-libvo-aacenc --enable-libvo-amrwbenc --enable-libvorbis --enable-libvpx --enable-libwavpack --enable-libx264 --enable-libxavs --enable-libxvid --enable-zlib
  libavutil      52. 63.101 / 52. 63.101
  libavcodec     55. 52.101 / 55. 52.101
  libavformat    55. 32.101 / 55. 32.101
  libavdevice    55.  9.100 / 55.  9.100
  libavfilter     4.  1.102 /  4.  1.102
  libswscale      2.  5.101 /  2.  5.101
  libswresample   0. 17.104 /  0. 17.104
  libpostproc    52.  3.100 / 52.  3.100
Input #0, mov,mp4,m4a,3gp,3g2,mj2, from 'yourfile.mp4':
  Metadata:
    major_brand     : mp42
    minor_version   : 0
    compatible_brands: mp42mp41
    creation_time   : 2015-12-22 23:09:46
  Duration: 00:05:27.04, start: 0.000000, bitrate: 5836 kb/s
    Stream #0:0(eng): Video: h264 (Main) (avc1 / 0x31637661), yuv420p(tv), 1280x720 [SAR 1:1 DAR 16:9], 5579 kb/s, 29.97 fps, 29.97 tbr, 30k tbn, 59.94 tbc (default)
    Metadata:
      creation_time   : 2015-12-22 23:09:46
      handler_name    : Alias Data Handler
    Stream #0:1(eng): Audio: aac (mp4a / 0x6134706D), 48000 Hz, stereo, fltp, 253 kb/s (default)
    Metadata:
      creation_time   : 2015-12-22 23:09:46
      handler_name    : Alias Data Handler

As you can see in the example above, my file contains two streams: the video stream (h264) as 0:0 and a single audio stream as 0:1

To get rid of the audio stream with ffmpeg, I simply needed to ask ffmpeg nicely to copy the file, keeping the 0:0 video stream, ignoring the audio stream and leaving the codecs alone (i.e., not trying to reencode anything):

ffmpeg -i yourfile.mp4 -map 0:0 -acodec copy -vcodec copy yourfile-silent.mp4

If you have multiple video streams or if you want to keep some audio streams, then just adapt the mappings accordingly.