The English word slave did in fact derive from the name for Slavs, but your friend's connection to the vikings is mistaken. The word thrall, as you note, was connected to the vikings, but both words probably entered the English language through Christian religious use, with the word slave first appearing well after the Viking Age (ca. 790-1060) had come to a close.

Native Old English had its own language for slavery, with þeow being perhaps most common term. Slavery had, however, softened somewhat prior to the Viking Age, with the þeow class of Anglo-Saxon England being afforded respect and rights that we might better understand as serfdom rather than slavery. This change might have happened early, since society was rather flat in Anglo-Saxon England after the collapse of Roman authority. Certainly, by the 700s there's little trace of any intensive use of slave labor.

But slave raiding was a key element of viking raiding beginning in the late 700s, and speakers of Old English needed a way to differentiate victims of exploitative slaving from their own servants tied to the land. They borrowed the Old Norse term þræll to help distinguish the victims such intensified forms of slaving. Variants on the word thrall first start showing up in English texts in the late 900s, so toward the end of the Viking Age. Some of the earliest uses are actually in biblical translations and commentaries, where churchman used the Norse term to describe the intense forms of Roman slavery recorded in the Bible, as opposed to the looser forms of servitude suggested by Old English terms for slavery. The borrowing of Old Norse þræll survives in modern English words like thrall and enthralled.

Meanwhile, on the opposite side of Europe, Slavs were being more frequently targeted for slaving. Arabic texts indicate that slaving was big business in Eastern Europe from at least the 800s, and they described the victims of this trade as Ṣaqāliba. This word is linked to the ethnonym Slav, but Arabic speakers had little interest or care in using the term precisely. More importantly, the Arabic term Ṣaqāliba never took on the meaning of "slave," nor was it borrowed into Western languages. From a linguistic perspective, that's a dead end.

From a European perspective, Slavs seem to have been increasingly targeted for slaving beginning around 1000, although some of this might just reflect the collapse of other slave markets, while the Slavs continued to be targeted. Current research shows that sometime during the 1000s, Latin writers around the Mediterranean began to use the term sclavus to distinguish a more rigid form of slave status from the broader and often less exploitative forms of servitude suggested by servus. This use slowly trickled across Western Europe.

According to the Oxford English Dictionary, the term slave first appeared in English in the late 1200s in the South English Legendary. Notably, this was a popular religious text, so it was written in a community where some people might have picked up the term on pilgrimage to Rome or the Holy Land. It was also a text that could be used for preaching, and so it might have helped spread the Mediterranean Latin term into the English vernacular. Other early entries in the Oxford English Dictionary are likewise poetic or religious, suggesting the term was used for its exotic flair rather than its precise meaning.

So there you have it. Thrall was in fact a borrowing from viking connections, but our evidence points to this being a strategic choice by English churchmen rather than an imposition from abroad. And English churchmen likely brought the word slave into the English language as well, whether through their travels to the Mediterranean or by their readings in Latin legal texts from the same place, but it's doubtful that many people early English users of the term slave recognized that this exotic new word had any connection to Eastern European peoples.