Etymology is not a determiner of whether something gets capitalized or not. Many proper nouns are derived from non-proper nouns (e.g. Apple, Smith, United Kingdom, World Wide Web), and vice versa (e.g. atlas, echo, narcissist, siren, sodomy). It is usage and usage alone that determines whether something gets capitalized or not.
This is decided for each and every word individually. And it is decided not by a committee in a single sitting on a given date, but by millions of people all over the world over the course of decades or even centuries.
As a result, there is no general pattern. The Roman Empire had nothing whatsoever to do with George Boole, after all. However, there can be a clear pattern for words of the same group, for obvious reasons. For example, the days of the week, the months, and the names of languages are always capitalized regardless of whether or not they were derived from proper or common nouns. Conversely, genericized trademarks (aspirin, bandaid, coke, escalator, kleenex, thermos, zipper) are no longer capitalized, even though they were specifically invented to be capitalized at all times ever.
Lastly, while capitalized nouns and adjectives are quite common in English, it simply does not like to capitalize verbs anywhere as much. So it is safe to say that even if we collectively try really hard to Google for Photoshopped cats in order to LOL, we will eventually end up googling for photoshopped cats and lolling.
See also: Should the word "Boolean" be capitalized?