SARS-CoV-2 variants are created by mutations in the virus's genetic code and not by authorities, as claimed in claims made on social media.
The theory was put forward in a Facebook photo captioned: “Let’s play spin the wheel of variants” next to a woman spinning a wheel of chance. This refers to location-related terms commonly used to label variants, such as “Indian variant”, “South African variant” or “UK/Kent variant”.
https://www.facebook.com/photo?fbid=541054710595737&set=gm.844323139799558
However, the Facebook user who shared the photo continues to mock people, calling them “stupid” if they believe the “stories” that there are variants of the coronavirus. He goes on to say in the comments: “We will continue to have the Japanese variety [sic] spread by eating fish.”
In a fact-checking assessment, Reuters found that, like all viruses, they naturally mutate over time, and SARS-CoV-2, the virus that causes COVID-19, is no exception (here). Changes in the virus's genetic code are mostly so small that they have little impact, but sometimes mutations can have worrying consequences: for example, they are more transmissible.
The location-related terms used to refer to these variants come from the name of the place where the mutation was first discovered. It is not a random selection process carried out by authorities.
https://www.nature.com/articles/d41586-021-01483-0
A list on the official UK government website details which variants are still under investigation and which are considered of concern. This list also provides numerous names for the mutations (including information on geographical origin), explaining that there is no internationally agreed naming system (here).