The blue stripes are intended to symbolize the stripes on a tallit, the traditional Jewish prayer shawl. The portrayal of a Star of David on the flag of the State of Israel was a natural choice, as it is a widely-acknowledged symbol of Judaism.
The Israelites used a blue coloured dye called tekhelet; this dye is now believed to have been made from the snail Murex trunculus. This dye was very important in both Jewish and non-Jewish cultures of this time, and was used by royalty and the upper class in dyeing their clothing, sheets, curtains, etc.