You can use the Haversine formula in conjunction with basic trig to iterate a series of vertices describing your circle.
Alternatively, if you have access to a GIS (e.g. ArcGIS, QGIS, PostGIS etc) or a GIS API (e.g. OGR, Shapely, GeoTools) you could simply buffer the point by one mile. E.g. for PostGIS you could use ST_Distance_sphere or ST_Distance_spheroid.
Also, one mile is not a huge distance so the error between spherical geometry and a planar approximation will not be great and may not be significant (depending on your use-case and accuracy requirements). So, you could consider working in a projected coordinate system (choosing a suitable one centered close to your area of interest will minimise error) and dispense with Haversine. Obviously over a greater distance it is more of an issue. Only you can say how critical it will be.