|tomato, Solanum lycopersicum|
|apple tree, Malus domestica|
Common names have no rules. I can call it Osage orange and you can say bodark, and we're both right.
We can call two plants by the same common name--marigold for both the plants below.
|marigold, genus Calendula|
|marigold, genus Tagetes|
Of course, synonyms and duplicate names create confusion. That's why scientific names were created.
There are no rules for writing common names either. Modern American English doesn't capitalize much of anything and that includes plant common names. If the common name includes a proper noun, such as Canada thistle and Queen Anne's lace, then, yes, a capital is called for, but only for the person or place, not the whole plant name. That is a very, very strong custom in North America. But not a rule. English classes teach us rules of grammar and what to do with plant names generally is not specified.
|Canada thistle, Cirsium arvense|
I looked at other sites in the English-speaking world: in Canada, in England, in India, across Australia. This custom seems to be confined to the Royal Botanic Garden in Melbourne, Victoria, Australia.
But, truly, there is no rule on common names. Custom only. Animal names are not generally capitalized. Birds are the exception. Bird-watchers have chosen a single, official, common name for each bird, and they capitalize those. Even that is not recognized by other fields; in my general ecology text books, for example, bird names (red crossbill, cattle egret) were not capitalized.
There are rules for English. Modern English does not capitalize nouns except for proper nouns, which are persons, places, or things. Another website defined proper nouns as being one-of-a-kind. So I'd say current usage implies that a whole plant or animal species is too many different individuals to be capitalized. You would of course capitalize an animal or plant individual name: the dog Lassie or General Sherman, one of the giant sequoias, or my Christmas cactus, Junior. But that is as close as the rules get, as far as I can tell.
|A Christmas cactus named Junior|
My advice: don't capitalize common names.
But, I agree with Tim Entwisle of RBG Melbourne on this: whatever you do, be consistent.
More on common names from this blog link
Spencer, R., R. Cross and R. Lumley. 2007. Plant Names: A Guide to Botanical Nomenclature. 3rd edition. CSIRO Publishing, Melbourne, Australia. link to their naming rules
Why the Christmas cactus is called Junior link