Where does モテ come from? モテる to be popular with opposite sex.
I've been following this thread with some interest. So far, I think folks are missing some of the underlying sense of 持つ, and the semantic overlap with certain English expressions that might more clearly explain the development to English readers.
持つ does include senses of "to hold", as mentioned by the original poster.
However, it also includes senses of "to get, to receive", and this is where we can start to see relevance to the "popular" sense at issue in the question.
Think about the English expression "to get some", as pertains to men and women (or more broadly, to any potential coupling scenario). The English phrase is specifically about getting (usually very particular) attention. The Japanese verb モテる is 持つ conjugated in the potential form, and thus the meaning is more like "to be able to get some", that is, "to be popular (with the opposite sex)".
(The above is based on limited research. If anyone has any backing from historical documents that could further elucidate the sense development, by all means, please post.)
モテる derives from 持てる. It is written with katakana to show its colloquial meaning. (Note that it is モテる and not *モてる. Similarly one has キレる, イケる, ウケる, etc.)
For example 大辞林 has
（動 タ 下一）
① 人気があって、ちやほやされる。 「女に－・てる男」
② 長くその状態を保つ。維持する。もちこたえる。 「共通の話題がなくて座が－・てない」
I guess the meaning of モテる can be explained as follows. 持てる means "to be able to carry/bear" and a secondary meaning is "to sustain/support". The meaning of モテる "to be popular" now derives from this extended meaning "to support/endorse". (Note also 持て囃す・持て栄す "to praise".)
Grammatically, this might be a bit confusing as the role of もてる is somewhere between active voice ("to support") and passive voice ("to be supported"), but on this point even the dictionaries cannot agree: For example 大辞泉 lists 維持される as a meaning, whereas 大辞林 (above) has 維持する.