Abarim Publications' online Biblical Greek Dictionary
Officially the words και (kai), meaning "and", and καινος (kainos), meaning "new", have nothing to do with each other but unofficially they obviously seem quite kindred.
The ubiquitous copulative particle και (kai) simply means "and" or "also". It's the most frequently occurring word form in the New Testament with 9224 appearances (the closest runner up is ο, o, the masculine singular definite article annex neutral relative pronoun, which occurs 3316 times). This comes down to 1.16 times per verse, so we understandably don't offer a concordance list for this word.
And και (kai) is all together not very exciting either. It's mostly used to tie words or statements together in much the same way as does our English word "and" but on occasion it's used to stress addition (Mark 10:26: who then can be saved?) or add emphasis (Matthew 8:33: they told all things and in particular of the demoniac).
Another particle of simple conjunction is τε (te), which does and means pretty much the same as και (kai) and even occurs in tandem with it. Whether this curious combination carries actual information or is rather cosmetic or dramatic of nature isn't quite clear, which in turn demonstrates that it probably is.
Together with the enclitic particle τοι (toi), which means "to you!", our word και (kai) forms the particle καιτοι (kaitoi), meaning something like "and you note!" or "and mind you" (Hebrews 4:3 only).
In additional combination with the enclitic particle γε (ge), which is a basic emphatic, the word καιτοιγε (kaitoige) means the same as καιτοι (kaitoi) but with added emphasis (John 4:2 and Acts 14:17 only).
Our particle και (kai) combined with the particle περ (per), which denotes entirety, forms the conjunction καιπερ (kaiper), meaning "though indeed" or "although". This word occurs 5 times; see full concordance.
Whether by convenient accident, design or a combination of both, the adjective καινος (kainos), means "new". The more familiar word for new is νεος (neos), which makes our adjective look like a combination of words that denote newness. The difference between καινος (kainos) and νεος (neos) is that the latter tends to express more of the same whereas the former expresses qualitative newness or novelty.
Our word may be used to denote newly produced items (wineskins: Matthew 9:17; garments: Luke 5:36), newly introduced teachings (Mark 1:27), newly issued commands (John 13:34) or new-and-improved things (a new song: Revelation 5:9; the new creation: 2 Peter 3:13). It all together appears 44 times; see full concordance.
- From our adjective comes the noun καινοτης (kainotes), meaning newness or renewal (Romans 6:4 and 7:6 only).
- Combined with the particle εν (en), meaning in or at, our adjective forms the plural noun εγκαινια (egkainia), meaning dedication (John 10:22 only).
The verb καινιζω (kanizo), meaning to renew, isn't used on its own in the New Testament, but combined with εν (en) it forms the verb εγκαινιζω (egkainizo), to dedicate or inaugurate (Hebrews 9:18 and 10:20 only).