Answer by taking a quote from the following article:

Clinton Richard Dawkins  (born 26 March 1941) is an English ethologist, evolutionary biologist, and author. He is an emeritus fellow of New College, Oxford, and was the University of Oxford's Professor for Public Understanding of Science from 1995 until 2008. Dawkins first came to prominence with his 1976 book The Selfish Gene, which popularised the gene-centred view of evolution and introduced the term, meme. With his book The Extended Phenotype (1982), he introduced into evolutionary biology the influential concept that the phenotypic effects of a gene are not necessarily limited to an organism's body, but can stretch far into the environment.

In his book, The Selfish Gene, Dawkins coined the word meme (the behavioural equivalent of a gene) as a way to encourage readers to think about how Darwinian principles might be extended beyond the realm of genes. It was intended as an extension of his "replicators" argument, but it took on a life of its own in the hands of other authors, such as Daniel Dennett and Susan Blackmore. These popularisations then led to the emergence of memetics, a field from which Dawkins has distanced himself.  Dawkins's meme refers to any cultural entity that an observer might consider a replicator of a certain idea or set of ideas. He hypothesised that people could view many cultural entities as capable of such replication, generally through communication and contact with humans, who have evolved as efficient (although not perfect) copiers of information and behaviour. Because memes are not always copied perfectly, they might become refined, combined, or otherwise modified with other ideas; this results in new memes, which may themselves prove more or less efficient replicators than their predecessors, thus providing a framework for a hypothesis of cultural evolution based on memes, a notion that is analogous to the theory of biological evolution based on genes.  Although Dawkins invented the term meme, he has not claimed that the idea was entirely novel, and there have been other expressions for similar ideas in the past. For instance, John Laurent has suggested that the term may have derived from the work of the little-known German biologist Richard Semon. In 1904, Semon published Die Mneme (which appeared in English in 1924 as The Mneme). This book discusses the cultural transmission of experiences, with insights parallel to those of Dawkins. Laurent also found the term, mneme, used in Maurice Maeterlinck's The Life of the White Ant (1926), and has highlighted the similarities to Dawkins's concept. James Gleick describes Dawkins's concept of the meme as "his most famous memorable invention, far more influential than his selfish genes or his later proselytising against religiosity".

Has anyone else been credited with coining "meme"?
Laurent also found the term, mneme, used in Maurice Maeterlinck's The Life of the White Ant (1926), and has highlighted the similarities to Dawkins's concept.