If you want to use one microphone to pick up the signal, it can be a very difficult task. Depending on how much the user can change the parameters, I can give a few ideas.
You can possibly use a uni-directional microphone and find a way to pinpoint the direction of the microphone towards the person you are trying to listen to, then use a bandpass filter with a bandwidth of about 300Hz to 3kHz (typical frequency range of the human voice). That will filter out any other noises that are of higher or lower frequency that aren't useful to what you're trying to capture. Of course, if there are other people in the path or if there are other sound sources within that 300Hz to 3kHz frequency, it can be very screwy.
300Hz to 3kHz is a very wide range, and encompasses almost everyone's range of vocal frequency. If the user has access to the filter parameters, a another solution would be to increase the Q-factor to narrow the bandwidth, and also be able to change the center frequency of the bandpass filter to adjust closer to the person's frequency range.
Then there's an issue of distance. If you want to use a standard headphones microphone or the microphone on a smartphone, then you will be very limited in what you can do.
Another way to do this would be similar to how the a person can create music tracks the isolate the vocal track of a song (a.k.a. an acapella). Basically, if the record company releases a "instrumental only" version of a specific song, a person can take the version, reverse the polarity of it, and combine it with the original song that has the vocals. This will cause the instruments to cancel with destructive interference, while the voice remains.
The same idea can apply here IF you somehow have the background noise WITHOUT the vocals you're looking for. I find this unlikely, unless you plan on using the setup for a staged scene that has an expected and consistent background noise each and every time you want to record the intended person's voice.
These are just a few suggestions and ideas that are off the top of my head. Like I said, this can be a very difficult task, and it can be more or less difficult depending on your knowledge and experience in audio and digital signal processing and frequency spectrum analysis.