With FFmpeg version 1.1 comes the option of high-quality audio resampling using The SoX Resampler library ('libsoxr').
To build FFmpeg with libsoxr, it must first be installed. On unix-like systems, it may be available as an installable package from your OS provider; otherwise, libsoxr should be installed from source code downloaded from here—either a versioned tar-file, or a git 'clone'. Once downloaded (and unpacked, in the case of a tar file), it should be built and installed according to the instructions in the INSTALL file (included in the source distribution).
With libsoxr installed,
./configure --enable-libsoxr instructs the FFmpeg build system to use libsoxr.
To invoke ffmpeg with the SoX Resampler, the option
-af aresample=resampler=soxr should be given; e.g.
ffmpeg -i my-48000.wav -af aresample=resampler=soxr -ar 44100 my-44100.wav
Further options may be given to change the default resampling bit precision and preserved bandwidth—see the manual page ffmpeg-resampler for details.