The question is why 'semicircle' is more popular than other variants, and, more broadly, why do any words with those prefixes have a different preferred form, even though they mean the same.
Since no one has apparently followed up on FumbleFingers' link or gave an answer, I'll summarize the answer here.
The last proposed factor seems to be the most relevant: the origin of the prefix and the rest of the word.
Demi- is the youngest of the three. It comes from old French/Norman and found its way into English along with many other Norman words. Demi was initially written separately, and was seen as an adjective or a noun, not a prefix. It was generally combined with words that have a similar root in French, or words in fields dominated by the French language, such as heraldry, clothing, or arms. However, the Oxford English Dictionary states that, as a prefix it is:
"Used with the senses ‘half, half-sized, partial(ly), curtailed, inferior’."
Looking at the OED, it seems other uses, such as 'demi-circle' stopped being used halfway through the 19th century.
An interesting definition, and, in a way, another difference with semi and hemi, is this:
"With ordinary class-nouns, indicating a person or thing which has half the characteristics connoted by the name; or is half this and half not, half-and-half; hence sometimes with sense ‘of equivocal quality or character’"
So 'demigod', but 'demi-beast', 'demi-Christian', and—my personal favourite— 'demi-Moor'.
On demi-circle, which appears to have been used a little in the 17th and 18th century, it says:
"A semicircle. Now rare"
Hemi- comes from Greek, and is therefore the oldest of the three. It is most often prefixed to other words that stem from Greek. This is most visible in the fields of geometry and chemistry.
The OED has this to add:
"Several Greek words containing this element were in use as technical terms in later Latin, e.g. hēmicyclium , hēmisphærium [...] In the modern languages they are very numerous, not only in terms adopted or adapted < Greek (directly or through Latin), but in new formations, scientific or technical, < Greek, or on Greek analogies. Words formed < Latin have the corresponding prefix semi- prefix; but there are instances of hybridism in the use of both prefixes"
It also provides this definition, distinct from the other prefixes:
"Half-; one half, the half, pertaining to or affecting one half; esp. in Anat., Biol., and Pathol. Pertaining to one of the two halves (right and left) of the body, or of any of its symmetrical organs."
On hemicircle, which was used for a bit in the early 17th century, the OED says:
"Obs. or arch. A half circle, semicircle."
Semi- comes from Latin. It is a rather direct translation of the Greek hemi- prefix, and is used in very similar cases (in Latin). Many Greek words containing hemi- were Latinized into semi. As Latin became the European language of science, semi- was often used in medicine and other sciences. The OED explains that the first use of the word was in compounds taken directly from Latin. It continues:
"In the 16th–18th c., the number of permanent compounds was increased mainly by the accession of terms more or less technical (many of them adapted or imitated from Latin), such as semibreve, semicircle, or semivowel. At the same time there was gradual enlargement of the scope of the prefix in the formation of general nonce-compounds, which became very frequent in the 19th c."
It also appears that 'semigod' in fact predates the use of 'demigod' by half a century, and that it was more popular in the 15th-17th century. Both are used to render the Latin 'semideus'.
To sum it up: there are actually many compounds of which multiple forms have existed simultaneously or consecutively. But most compounds today have one dominant prefix, largely influenced by usage in science and engineering. Whenever there are multiple forms still in use, they usually have very distinct meanings.
So while a 'semicolon' is a punctuation mark, used perhaps too little, a 'hemicolon' is the left or right half of the colon (i.e. the end part of the large intestine), and is probably used sufficiently ;)
