This is a theory that has found some very limited support in the academy. Not a translation of the Hebrew Bible, or the New Testament (remember that Jesus is the prophet named the most times in Quran) but a Syriac language lectionary. Honestly, the Wikipedia article on The Syro-Aramaic Reading of the Koran: A Contribution to the Decoding of the Language of the Koran by Christoph Luxenberg (a pseudonym) better than I could.

Is it a convincing argument? I'm not a philologist, and I know no more than a shmattering of Arabaic, but I think the article is convincing that there might be a heretofore mostly unrecognized influence of Syriac on early Arabic and suggestive that maybe some of the hard places in the Qu'ran might be easier if we compare them to Syriac vocabulary and idioms (notice the mighty and maybes). However, the argument that we should mostly ignore indigenous Muslim commentaries on the Qu'ran and that the Qu'ran is "based on" Christian Syriac lectionaries (rather than influenced by their language) is much, much less convincing. (Edit: I should make it clear that no one else has really bought his thesis, either. Literally, I don't know of a single academic. It's at most "interesting" and "thought-provoking", but nowhere near "convincing" or "right". And most specialists in the area probably wouldn't even call it "interesting"--see Wikipedia for the scathing criticisms from other academics).

It's worth noting that there is at least one story (Abraham smashing his father's idols, and then pretending it was one of the idols that did the smashing) in the Qu'ran that, to my knowledge, doesn't show up in Christian sources, doesn't show up in the New Testament or the Hebrew Bible, but does show up in the Talmud. This suggests to me that, if we bracket the claims that the Qu'ran was divinely given, we can see the influence of multiple traditions on the Qu'ran and the early Muslim community. But "plagiarism" doesn't really apply. That's not really a term that makes sense in this context. Rather, it's clear that the early Muslims saw themselves as continuing (and purifying) the traditions already found among other "Peoples of the Book". Indeed, it's always stuck me as interesting how much the Qu'ran assumes a somewhat knowledgeable audience--one already familiar with a lot of the Jewish and Christian stories it makes references to. This extra-Quranic tradition continues to be important, for instance, while it's universally accepted in Islam that 'Adam's wife was Hawwah (Eve, cf the Hebrew Havva), even though that name does not appear in the Qu'ran.

Interestingly, this is an allegation that the Qur'an itself refutes.

Qur'an 25:4-6: "And those who disbelieve say: This is nothing but a lie, which he has forged, and other people have helped him at it. So indeed they have brought an iniquity and a falsehood. And they say: Stories of the ancients, which he has got written, so they are read out to him morning and evening! Say: He has revealed it, who knows the secret of the heavens and earth. Surely He is ever forgiving, merciful."

This suggests two interesting things. Firstly, it points to the fact that many of the stories found in the Qur'an were commonly known in the community and time period that the Qur'an developed, enough that they could be recognized and called "Stories of the ancients". Secondly, it suggests that this allegation, that the Qur'an was just either just made up by Muhammad ("this is nothing but a lie") or, as you suggest, "plagiarized" from older stories, was being used against the early Muslim community. It certainly is likely that oral, and perhaps some written, versions of those stories were circulating in the community at the time.

It would be unlikely though that the narrative stories in the Qur'an were directly copied from other sources, however. The Qur'an contains stories found in the old and new testament but differs on a lot of details, and contains stories not found in those sources. Take for example the account of Mary in the chapter of the Qur'an named for her, Sura 19. The story of the annunciation is found in Qur'an 19:17-21:

"Then We sent to her Our spirit and it appeared to her as a well-made man. She said: I flee for refuge from thee to the Beneficent, if thou art one guarding against evil. He said: I am only bearer of a message of thy Lord: That I will give thee a pure boy. She said: How can I have a son and no mortal has yet touched me, nor have I been unchaste? He said: So it will be. Thy Lord says: It is easy for Me, and that We may make him a sign to men and a mercy from Us."

This account is can be compared to the annunciation story in the first chapter of the gospel of Luke. It's very similar, but not the same: in Luke, the Angel Gabriel is explicitly mentioned to have visited Mary, not the holy spirit or some other representative, and the wordings of their conversation is different; notably, the angel tells Mary that her son will be called the Son of God, a declaration absent from the Qur'anic version (for good reason, since Jesus is not a divine being in Islam). This is followed, however, by a different story of Mary in Qur'an 19:22-25:

"Then she conceived him, and withdrew with him to a remote place. And the throes of childbirth drove her to the trunk of a palm-tree. She said: Oh, would that I had died before this, and had been a thing quite forgotten! So a voice came to her from beneath her: Grieve not, surely thy Lord has provided a stream beneath thee. And shake towards thee the trunk of the palm-tree, it will drop on thee fresh ripe dates."

This is a story that isn't from Luke, in fact, it's not in any of the canon books of the new testament but is found in an apocryphal gospel called the "Infancy Gospel of Matthew", though in in a slightly different form. In this version, the infant Jesus commands the palm tree to bend over, which it does, so that Mary and Joseph can gather its fruit. There are other apocryphal stories in the Qur'an too, such as an account of the infant Jesus talking while in the cradle.

This is an example from only one chapter of the Qur'an, but I think it illustrates the point well. If the Qur'an were more of a one-to-one copy of earlier writings it would be more apparent. More likely, the Qur'anic accounts are based on oral traditions of these stories that were circulating at the time.