Before there were balls called "wiffle balls," children would have played stick ball with a stick and whatever sort of soft ball they could come up with -- perhaps a tennis ball, perhaps a rolled up pair of socks.
The original "wiffle ball" was created by David N. Mullany, who had played baseball for the University of Connecticut. He wanted something his 12-year old son (David A. Mulaney) and friends could play with that would cooperate with trying to pitch a curve ball. He created balls using round plastic packaging materials from perfume bottles. His best-working design featured a solid side on half the ball and holes carved out on the other half of the ball. Mullany began the Wiffle Ball company, which was later taken over by his grandsons.
For more details, check out "History of the Wiffle Ball" by Marti Attoun on American Profile (June 5, 2005).