If you got a good signal from the interface box, the problem is on the house side of the box not on the outside Verizon lines. The next time it rains and things get noisy, unplug everything that is inside. This includes cordless base units, fax machines and modems. Plug one phone in and check. If it is noisy try another phone in the same jack. Still noisy? It's a wiring problem. Good signal? Add instruments one at a time and check at each position. If it turns out to be a phone, go shopping for a replacement. If it is everywhere with any phone, call Verizon back, show them what you have done and recorded on paper and tell the repair guy to fix it. If you have the plan that pays for inside repair, you are in god position. If you have to pay for wiring service you might try to locate the problem yourself or enlist a savvy friend.
Good luck!
Charlie