Renumeration vs Remuneration (reimbursed financially), which is correct?
Solution 1:
According to the OED renumeration / remuneration are interchangeable. So too are the associated verbs - renumerate / remunerate. However, some commentators have strong feelings about renumeration being used with its first-cited meaning (i.e. remuneration, see below) "... to be avoided at all costs is the metathesized form renumeration." (R. W. Burchfield New Fowler's Mod. Eng. Usage (1996) 666/2). Remuneration is much more commonly used than renumeration (https://goo.gl/44GQRa).
Remuneration is first cited c.1400, comes from Middle French (remuneracion) and Latin (remūnerātiō), and means reward, recompense; (now usually) money paid for work or a service; payment, pay.
Continuously since the mid 16th century renumeration / renumerate have been used with the same meaning as remuneration / remunerate.
Since 1596, renumeration has also meant 'the action or an instance of numbering or enumerating again'. This is widely held to be it's primary meaning.
The OED suggests that renumeration with the meaning of remuneration is a variant by metathesis (https://goo.gl/lhMZQh) arising from a folk-etymological association with words in numer-.
Solution 2:
Well, my dictionary defines renumeration as "Misspelling of remuneration".
Solution 3:
When I started looking into this, I absolutely thought it was renumeration. Turns out Remuneration is WAY more popular. The OED website has the first recorded use of Remuneration at circa 1400 and of Renumeration over 150 years later. There is noted controversy over the use of "Remuneration", but only from a single source, listed by @Dan above.
Given that "remuneration" is used more than a 100 times more frequently in literature than "renumeration" (according to Ngram, linked above), I would say that answers your question. It also has a more logical etymological root, coming from the Latin remūnerātiō, meaning the same. "Renumeration" appears to lack any etymological root other than "probably formed via metathesis from "Remuneration".
It's going to take an awful long time for me to get used to it though...