First of all, while Friday is considered sacred to Muslims, it's not a Sabbath in a sense that "thou shalt not work"; in terms of religious practice, the only thing that separates Fridays from all the other days is the congregational prayer that all men are obligated to attend at noon. After that, you may return to your work.

#Did the specific day of the week commemorate some other event?

Friday prayers are mandated in the Qur'an:

O you who have believed, when [the adhan] is called for the prayer on the day of Jumu'ah [Friday], then proceed to the remembrance of Allah and leave trade. That is better for you, if you only knew.

And to draw the line in the difference between the Jewish Sabbath and the Friday prayers, in the next verse:

And when the prayer has been concluded, disperse within the land and seek from the bounty of Allah , and remember Allah often that you may succeed.

The above is from the Qur'an, 62:8-9. Incidentally, this verse belongs to the chapter known as Al-Jumu'ah (Friday).

And finally for the Shi'ites, the Battle of Karbala takes place on a Friday; it is a mournful event that is commemorated annually.

I originally typed this up to respond to someone with a really poor answer, but here is the basic gist of the Christian position.

Orthodox, Catholic and many Protestant denominations still affirm that Saturday is the Sabbath (for ex, Luther saw Sunday as a civic institution, not as a new or different holy Sabbath day). Later low-church expressions of Protestantism claim that Sunday is a new sabbath from a kind of antinomian position, but that is a late innovation. Before that, we get commentators like Ignatius and later Origin merely commenting on the fact that Sunday has become "like" a new Sabbath without making the claim that is has in fact replaced the Sabbath.

In a very general sense, the transition from the observance of the Saturday Jewish Sabbath goes back to Mark 2:27 ("the Sabbath was made for man, not man for the Sabbath") and as part of Paul's program of delimiting aspects of the Jewish law which are not obligatory for Christians (Colossians 2:16; Galatians 4:9-10; Romans 14:5). These weakened or removed the obligations Christians had to observe on Saturday.

Theologically, worship on the first day, Sunday, has its basis in Acts of the Apostles (20:7) and Paul's letters (1 Corinthians 16:2), but in those cases it's referred to as "the first day," not as a new Sabbath. It is also mentioned in the Didache, a document dated somewhere near the end of the first century/beginning of the second, where it is referred to as "The Lord's Day."

The patristic authors actually referred to Sunday as "the eighth day" and connected it more to the day of Christ's resurrection, and to the notion of "a new creation" commencing after the seventh day of God's creation, than the Jewish Sabbath as such. This is basically the earliest explanation we have for why Christians chose the first/eighth day, but it's significantly later than the practice began, so it very well may not be a reliable insight into why the Christians of the first century chose that day. Even still, the early church probably celebrated both the Sabbath and the eighth day (or Lord's Day) together for some time. The observance of Sunday alone came gradually over the first few centuries of Christianity, rather than after some specific decision or ruling.

The distinction is basically that Saturday is "the" de jure Sabbath for Christians--but it doesn't hold any obligations--while Sunday is "a" de facto sabbath by virtue of the fact that Christians happen to worship on that day.

Is this combination of holy days where the notion of having weekends off first came from?