|Recent stamp commemorating al-Khwārizmī|
Perhaps I should say Classical origins, since the Babylonians developed an arithmetical system for dealing with linear and quadratic equations. The Greeks, Chinese, and Egyptians used a kind of geometric algebra in the centuries BCE. A Greek mathematician in Alexandria in the 3rd century CE, Diophantus, is sometimes called the "Father of Algebra" based on his series of books, Arithmetica, that deal with solving algebraic equations.
Diophantus has a rival for that title, however.
An Arab mathematician named Muhammad ibn-Mūsā al-Khwārizmī (c.780-850) wrote a book in Baghdad in 825 called Kitab al-jabr wal-muqubala ["The book of restoration and balancing"]. Specifically, the process of turning the equation x - 2 = 12 to x = 14 was called jabr because one was "restoring" the x. The process of turning x + y = y + 7 into x = 7 was muqubala because one was "balancing" the two sides.
The word al-jabr, "restoration," eventually became the sole label for this method of mathematics.*
A Latin translation of his work was circulating in Europe in the 12th century. Fibonnaci is believed to have been exposed to Arabic mathematics, which might be why he was able to come close to solving the equation x3 + 2x2 + cx = d.
So al-Khwārizmī gets the title "Father of Algebra" because the branch of mathematics is named for his book describing it. He also gets the honor of naming a different mathematical term: his name was Latinized into Algorithmus, from which we derive the term "algorithm."
*Interestingly, this word's non-mathematical definition of "restoration" made it suitable for other uses. It made its way into European parlance via Arabic, and "algebrista" became a title for a "bone-setter." The term could also apply to barbers, because they did bone-setting as well as blood-letting. (The term for a blood-letter was "sangrador.")