Our hosts, yourdictionary.com, give the origin of "tide" as Old English "tid", meaning a division of time. You can still see this in Christmastide, and in the word for "time" in other Germanic languages such as Swedish (also "tid"), Dutch ("tijd") and German ("Zeit").
For "tide over", the example they give conveys the meaning perfectly: "I asked for $100 to tide me over till payday."
However, IMO the definition they give ("To support through a difficult period") is rather misleading. It’s not the support or the difficulty that’s important, it’s the period. The $100 provides a temporary solution that gets you through the time until a proper solution can be found.
Ed