Variable stars were for a long time thought to be rare, so that the naming system only evolved as more and more were found. If a variable already had a Bayer letter, then this was used and no extra designation given. Examples are Delta Cephei, Beta Lyrae, & Beta Persei. An added complication is that ancient proper names, mostly of Arabic derivation, are still used even now for many bright stars. Hence Alpha Orionis, which is slightly variable, is more often called Betelgeuse, and Omicron Ceti is Mira - a name which also stands for the class of Long Period Variables of which Mira was the first to be found by D Fabricius in 1596.
After 1850, the great Durchmusterung surveys (Bonner Durchmusterung {BD} North & South, Cordoba Durchmusterung {CD}, and Cape Photographic Durchmusterung) catalogued hundreds of thousands of stars and led to many new discoveries. Argelander, the initiator of the BD, believed that there would be few variables and brought in a system of naming them with the last 9 remaining capital letters of the alphabet (R to Z) followed by the Latin genitive version of the constellation name.
As more variables were discovered, double letters were used starting RR, RS, RT,... RZ, followed by SS, ST, SU,... SZ, and so on upto YY, YZ, ZZ. Still these were insufficient, particularly as photographic techniques accelerated discoveries, so the series continues with AA, AB, AC,... AZ, BB, BC, BD,... BZ, ending with QQ, QR,... QZ. (the combinations starting RR having already been used.) The letter J is not used nor reversed order pairs, so 334 combinations are possible for each constellation.
A A Nijland, a Dutch astronomer active around the turn of the 20th century,
proposed a more sensible system of simply numbering variables, V1, V2, etc.,
but by then the letter system was in widespread use and variable names like
U Gem, SS Cyg, and RR Lyr, were being used to denote
types
of variability, in the same way as Mira.
However, Nijland's system is used after the series of
letters, so the next variable after QZ is V335, then V336, and so on.
The BAAVSS memoirs, now largely incorporated into
the computer database, contain many of Professor Nijland's observations.
This is the system used in the General Catalogue of Variable Stars (GCVS) and agreed internationally. However, many stars become suspected of variability and are catalogued numerically in an extension to the GCVS called the New Suspect Variable catalogue (NSV). Formerly there was the Catalogue of Suspect Variables (CSV), and some individual observers and organisations list their own finds with various temporary designations. Novae and supernovae also have initial discovery names that remain in use after they receive an official variable star designation. For example GK Per is still referred to as Nova Persei 1901.
Some confusion may still arise with new discoveries and suspected variables, but the evolved system described above is standard for confirmed variables.