We don't name stars that way, but there are two other stars with traditional names starting with Rigel, and the closer one is in about the right place. This is Mu Virginis, Rigel Al Awwa, so I've picked that one for the "near" Rigel.
Now this star has the official name of Almach according to the IAU, but it is a multiple star system and the name only applies to the primary star. Sometimes other stars in the system are given alternate traditional names for the system (Rigel Kentaurus and Toliman, for example) so it's possible that Gamma-2 Andromedae would be given this alternate name.
Mentioned in TOS as Deneb II and V. Star Charts calls this Deneb Kaitos so I went with that.