I've done a lot of research on Syriac or Aramaic. Syriac is the Greek invention for all who wrote Aramaic. So, Syriac is not the same as 'the language of Syria.
Unfortunately, there has been a historical genocide on all Syriac Bibles during the ages, especially by the Western Churches on all the churches that did not agree with the chalcedon counsel.
They are the Churces of the East.
So, we live with a Christian myth, that christianity spread itself in the Roman empire, however, Christianity already reached China in 635 AD!
That was even before they reached the Netherlands in the 9th century by 'Bonifatius' :)
In the 19th century, China made known to the world, about a discovered monument which contains all the names of the missionaries who did all the hard work.
I got a small part of the Hsian-Fu monument, but you get the idea!

Why do I write this? Because I wanted to show that Syriac, was up to the Islam age, a WORLD-language.
It was even used by the Parthens, who owned e.g. Babylon.
According to Peters letters, he stayed in Babylon. Babylon has -never- been part of the Roman empire, neither Babylon was in 'decline' (as the WTS seems to teach :) )
From Elam/Parthia/Mesapotamia Syria, up to Judea, the -main- language was Aramaic, not Jewish, not Greek.
This was the Parthian Empire during Jesus age.

Of course, there were Dialects. So, Jesus, seemed to have addressed God with 'el' (el-i means My God, it's the genitive)
While in Judea, alaha was the way to 'address' God.
And some may have pronounced it as 'aloha'
Also Peter (Cefas) was found to be 'galilean', not because he spoke Greek, but because of his other Aramaic accent!
Even Paul, seems to have spoken Aramaic. Wasn't he speaking 'hebrew'? Perhaps, but Jews, were addressed as 'jews', not as Arameans, they were addressed as speaking 'hebrew' while they in fact even -might- spoke (+/- 50 years ago) Jiddish or another language.
And, even the jews in Babylon, wrote the Talmud, which is in Aramaic language, not the 'Hebrew' language of the OT.
So, this was a short historical backup!
Is there any backup that the NT was written in Aramaic? Sure!
If you compare the Greek variances (there are sometimes 4 variances!) they ALL are explained by looking at the Syriac words, not the other way around.
Even Irineaus has made such remark.
Again, Syriac is NOT just the Bible in Syrian territory. Syriac is the Greek name for ALL aramaic documents, written using 'Estrangelo' script and some variances.
b.t.w. Irenaus, did not say: Greek is the original, but he wrote about THE greek originals :)
When one looks at the Syriac Bible (I'm speaking about the Peshitta/Peshitto) you will find lots of Biblical contradictions or 'spurious parts' As Jim calls it :), vanished and explained because some translator just did not get the Judean meaning of some proverb.
Just one example (I've already collected +/- 500 differences)
According to Jesus, the Farisees were praying 'as if' and they consumed the widows houses!
What's the relation between praying and consuming the houses? I think, that a biblical commentator would say: "Well, why pray to god, while consuming a widows house?"
(Matthew 23:13 AND luke 20:47)
But the Aramaic word were greek for 'pretext' has , has several meanings, and the word can mean 'offer' or 'offerings'.
So, the Pharisees, asked money for praying long prayers, and so they consumed the houses of widows.
Wasn't that pracise a former middle-age Catholic practise as well? Now Jesus words, start to have meaning! Just because of looking at the Aramaic source
Of course, it is possible that some letters, were 'both' originally Greek as Aramaic. Take Galatians and 1 Peter. The Greek grammar seems to be ok but so does Aramaic. And the Galatians _were_ Greek speaking, not an 'Aramaic' people.
But when we find other letters of Paul, we find 'maran-atha', 'Abba' in the Greek text, while the Syriac version, exactly has THOSE words on respective places. So, which is the original?
When we compare Hebrews, which is a real difficult letter, the Syriac even has (my findings) +/- 40 little ot big differences which really make Paul a 'normal' or 'big' writer.