When I first began translating Jesus's words, I thought that he had invented many of the Greek words that he used because the first time they appeared was in the Gospels. However, my mistake arose from my use of the Perseus database as my resource for ancient Greek texts. That database doesn't include one key text, the Septuagint, the Greek Old Testament of Christ's era. I have since discovered that many of the words that seemed original to Jesus were first used, in one form or another in the Septuagint.
Closer study, however, demonstrates that Jesus clearly used the Greek Septuagint as the source of his word usage. If a Greek word appears nowhere else in ancient Greek literature, it likely appears there. There are a couple of exceptions (notably the mystery word), but for dozens of other uncommon words, it is clearly the source. When Jesus quotes the OT, he uses the Septuagint vocabulary and often the exact words of the work. He also sometimes uses paraphrases, but when he changes a word, we can clearly see that he does so for a reason.
Not an Artifact of Translation
Some may claim that this correspondence is an artifact of translation from the original Aramaic, but it is hard to imagine how. The probability of translated a given Aramaic quote into the exact same Greek of the Septuagint, down to every word form, case, and tense is zero. Aramaic word forms, especially the verb forms, are not close enough to those of Greek. For example, in English translation, we absolutely cannot tell the verb form of the word, even its tense, from English translation.
So, we might think that the translators of Jesus's Aramaic went to the Septuagint when Jesus is quoting the OT. This would explain the exact quotes, but it doesn't explain the paraphrases and plays on words. It makes them harder to explain. If one evangelist always quotes and another always paraphrases, it would also make some sense, but that is not the case. Quotes, paraphrases, and word plays occur randomly in the synoptic Gospels, but less so in John generally.
We do know that Jesus was not quoting the Hebrew Bible. Jesus often uses plays on the Greek words that simply do not exist in Aramaic. When Jesus does use Aramaic words, they often appear in the Septuagint first. See the more detailed discussion below. Jesus willl also often use a Greek word that is a common translation of many different Hebrew words.
The Nature of the Septuagint
The Septuagint was written about a hundred years before Jesus. Like our Bible became the source of many English words used today (and Latin words back in the fourth century), the Greek Septuagint was the likely source for almost all of the unusual Greek words Jesus uses. Almost all because there is one word (discussed here) that appears only in Jesus's words.
The First Record of Judean Greek
This doesn't mean that the Septuagint was the original source of these words, but simply that it is the oldest written source that we have of them. Some words, such as diallasso, which is translated as "be reconciled" appears only once in the NT (Matthew 5:24) and it also only appears once in the Septuagint (1Samuel 29:4), but these two verses have no relationship to each other except that both verses involved settling differences between opponents. From this, we can infer that the word was generally used for this purpose, but that it was rare. It was so rare that it doesn't even appear on Strong's Concordance. Interestingly, though both of these verses mention an "adversary" in English translation, the two Greek word that are translated as "adversary" are different.
The Septuagint documents the differences between Judean provincial Greek forms not commonly found in other forms of the language. For example, ταχὺ would normally be a neuter, singular adjective meaning "quick." However, Jesus uses this form three times to mean "quickly" when the proper Greek adverb form is ταχέως. Many might assume that this is an example of the "poor Greek" of the Bible writers, but the fact is that ταχὺ is the form used to translated the adverb quickly throughout the Septuagint starting with Genesis 27:20. This indicates that this was likely the adverbial form of the word used by the Judean people long before Jesus.
Sometime the differences aren't only a matter of form, but a matter of concept. For example, the Greek word for altar or shrine is βωμός, but the word used in the Gospels is θυσιαστηρίου, a word based on the word "sacrifice." This word only for the first time in the Septuagint.
In the case of one uncommon verb, skandalizo, the word doesn't appear in the Septuagint itself but it comes from another word that does. The verb is the source of our word "scandalize" and most Bible versions translated it as "offend." However, the word is from a noun in the Septuagint, skandolon, meaning "trap," "snare," or "stumbling block." This noun isn't used elsewhere in Greek, but it appears twenty-five times in the Septuagint. The noun is changed to a verb by adding the "izo" ending very much like we add "ize" to a noun in order to make it a verb. So, literally the word would mean to "stumblize." In English, we would simply say, "trips up" capturing the same idea exactly. From its real meaning, biblical translators interpret it more somberly to mean "to give offense" and "to scandalize."
As a Key to Research
For word that Jesus doesn't use very often, the Septuagint is a useful guide to help use get a more complete picture of how Jesus and the people of his time heard a word. One example is agalliao, ἀγαλλιᾶσθε, translated as be exceedingly glad. This word is used a number of time in the Septuagint, but it isn't used to translate one Hebrew word, but a number of different one. It is first used in Psa 2:11 , where it translates the Hebrew word giyl, which means "rejoice." In Psa 16:9 and Psa 32:11 it is used to translated samach. "be glad." In Psa 33:1, ranan, "cry out." In Psa 68:4, `alaz, "exult." In Psa 81:1 and Psa 98:4 , ruwa` "shout." In Isaiah 12:6, tsahal, "cry shrilly." So it basically means "shout for joy.