Q&A: Is the Phoenician alphabet suitable for writing modern Hebrew?

You could—the letters are substantially the same (functionally speaking) as the modern Hebrew alphabet—but it is graphically less sophisticated than its derivatives, namely the Greek & Roman alphabets, and the Assyrian script which gave rise to the revised form of Hebrew from the Second Temple period onwards, known today as “Square Script”:

Why does Greek have names for their letters such as alpha, beta, lamda?

The names of the Hebrew/Phoenician alphabet were given by the ingenious Canaanite slave(s) who first invented them some time in the 1800s BCE, possibly in Wadi El-Hol in Egypt:

Detail of an inscription on a rock face in Wadi El-Hol, Egypt (near the Valley of the Kings)