We have all dropped a rod or two overboard and have fished for them with treble hooks and deep diving cranks. Purchase a snap type stringer(about $10-$15). The keepers are strong enough to catch onto a rod and soft enough to straighten out when snagged on the bottom.
Tie one on a small rop (I have used it on a heavy rod with braid), open the keepers, sling it out over where your rod was lost and retrieve.