Mira B
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Long known to be erratically variable itself, its fluctuations seem to be related to its accretion of matter from Mira's stellar wind, which makes it a symbiotic star.
Orbit
Its orbit around Mira is poorly known; the most recent estimate listed in the Sixth Catalog of Orbits of Visual Binary Stars gives an orbital period of roughly 500 years, with a periastron around the year 2285. Assuming the distance in the Hipparcos catalog and orbit are correct, Mira A and B are separated by an average of 100 AU.
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Current research
In January 2007, astronomers at the Keck Observatory announced the discovery of a protoplanetary disk around Mira B. Discovered via infrared data, the disk is apparently derived from captured material from Mira itself; Mira B accretes as much as one percent of the matter lost by its primary. Though planetary formation is perhaps unlikely as long as the disk is in active accretion, it may proceed apace once Mira A completes its red giant phase and becomes a white dwarf remnant.
Several factors, such as low x-ray luminosity, suggest that Mira B is actually a normal main-sequence star of spectral type K and roughly 0.7 solar mass, rather than a white dwarf as first envisioned. However, a 2010 analysis of rapid optical brightness variations has indicated that Mira B is, in fact, a white dwarf.