A basionym is the original name of a taxon on which a new combination (or name at new rank) is based. This term and its concept is confusing! An original, or base name, is not always a basionym. For example, Nupela frezelli is not a basionym; basionym is undefined. However, a taxon that has been transferred to a new genus does have a basionym. For example, Navicula mutica is the basionym of Luticola mutica.
The International Code of Algae, Fungi, and Plants (ICN) is the set of rules that guides the scientific naming of algae (Turland et al. 2018). Earlier editions of the code were translated into Chinese, French, Italian, Japanese, Korean, Portuguese, Spanish, and Turkish. The English version is considered definitive and is available online.
The rules can seem imposing and dense, but the text is important for setting out a means for organisms to be named in a clear and meaningful manner. It is worth the time to understand the code, at least parts of it.
According to the glossary of the ICN, a basionym is
The legitimate, previously published name on which a new combination or name at new rank is based. The basionym does not itself have a basionym; it provides the final epithet, name, or stem of the new combination or name at new rank (Art. 6.10) (see also name at new rank, new combination).
Cite the Code:
Turland, N. J., Wiersema, J. H., Barrie, F. R., Greuter, W., Hawksworth, D. L., Herendeen, P. S., Knapp, S., Kusber, W.-H., Li, D.-Z., Marhold, K., May, T. W., McNeill, J., Monro, A. M., Prado, J., Price, M. J. & Smith, G. F. (eds.) 2018: International Code of Nomenclature for algae, fungi, and plants (Shenzhen Code) adopted by the Nineteenth International Botanical Congress Shenzhen, China, July 2017. Regnum Vegetabile 159. Glashütten: Koeltz Botanical Books.