Is there a connection between the word "nine" and "new"? The two words are similar in many languages.
As Colin Fine says, the words are somewhat similar in Proto-Indo-European: 'nine' is PIE *h1neun, 'new' is PIE *neuo-. The latter word seems clearly derived from *nu 'now': 'new' is 'that of now'. Phonologically it seems difficult to relate *h1neun to *nu/*neuo-, both because of the initial laryngeal and the final n, neither of which could be added by any regular derivational process; and semantically there's no obvious relationship between 'nine' and 'new'. So they're most likely unrelated - although see fdb's answer for some reasonable arguments to the contrary.
This is a long-standing discussion. The idea is that the ancestors of the Indo-Europeans counted on the four fingers (not including the thumb) of one hand. *oktō is a dual meaning “two spans”. The word for “nine” introduces a “new” set of four. The mentioned phonological objections are not insurmountable. The initial laryngeal has been posited mainly to explain the initial vowel in Gk. ἐννέα but this is not the only possible explanation (prefixed ἐν according to Schwyzer, Griech. Gr. I, 591). In principle you could reconstruct the PIE word for “nine” as *newṃ (Latin novem) and regard this as the neuter singular of *new- , with secondary assimilation of n…m to n….n in Germanic etc. This is all pretty much mainstream IE theory; of course, none of this is certain.