Tap Portugal is pricing Miami to Lisbon nonstop flights at $637 round trip. However, fly TAP Portugal to Luxembourg with a 5-night stopover in Lisbon and the price drops to $388 round trip.
TAP Portugal’s website asks if the flyer wants a stopover in either Lisbon or Porto on the outbound or return flight of a round trip ticket search from the USA. In several searches this year I have tried to add a stopover, but the price usually jumps hundreds of dollars. This is one of the first deals I have found to combine a long stopover in Lisbon with a cheap round trip ticket to another European city.
TAP Portugal has Miami to Luxembourg priced as low as $348 round trip without a stopover. In my search I found adding a 5-day stopover in Lisbon on the return only increased the ticket price $40.
Cheap TAP Portugal flights with a long stopover works for other cities too. Miami-London with 4 nights in Lisbon prices at $447.
TAP Portugal  Miami MIA – Lisbon LIS – Luxembourg LUX – Lisbon LIS (5 night stopover) – Miami
There are also $473 tickets Miami to Lisbon on Royal Air Maroc with potential 27 hour layover in Casablanca, Morocco in both directions.
Royal Air Maroc Miami – Casablanca (27 hours) – Lisbon – Casablanca (27 hours) – Miami