What size are the 4 people? A family of four adults will likely be kind of tight in a Mooney for any significant period of time (and a 1000 nm trip will take a significant amount of time). Two adults and two kids would be fine, at least until the kids grow up. Cargo/baggage space may also be an issue, particularly if you're planning on staying a long time.
Save money - Maybe, depending on how you work the numbers. If you count only direct operating costs (gas & oil), probably so. For a 2000 nm round trip, one fuel stop each way, you'll be burning about 150 gal. of fuel and flying about 14-15 hours. At $6/gal, that's $900 in gas. You might need a quart or two in oil over that time, but that cost would be minimal. If, instead, you add up all your ownership costs (parking, maintenance, insurance, carrying costs, etc.), and divide by the number of hours flown, you'll come up with a very different number.
Save time - quite possibly. You could make a 1000 nm trip in about 7.5 hours of air time, and you'd need a fuel stop. How long that fuel stop takes is largely up to you--30 minutes is probably a minimum, and you (and/or the family) may want to take a bit more time to stretch the legs, use the restroom, etc. Other variables would be the distance to the airport on each end, and the possibility of weather delays.
Flexibility - Definite yes, weather permitting.
For $80k, you could easily get a nice M20F, and probably a decent M20J. The J would be preferred, all other things being equal, due to being newer and faster on the same fuel burn (10-15 kts faster). You'd really want an IFR GPS (WAAS preferred) and a capable autopilot, and your family may have something to say about general appearance.
Other ownership costs are a definite issue. Some of the biggies are maintenance, insurance, parking (hangar or tie-down), and loan payments if you finance the plane. Mx is widely variable. I've had very little unscheduled Mx in the 4 years I've owned '48Q, maybe $2k worth. My last two annuals were under $1k each, but I was working on them with my A&P. My first two were $2k-3k unassisted. It's commonly recommended to set aside a fair portion of the purchase price (I've seen numbers from 5%-20%) of the plane for unexpected maintenance the first year. If you're mechanically inclined, you can do a lot of the routine maintenance on the plane yourself, and you can do anything with your A&P's approval and under his supervision.
Insurance will depend a lot on you--what ratings, how many hours (total, complex, and Mooney), any negative history. I paid around $1800 for my first year (4 years ago) with 80 hours, no instrument rating, no appreciable complex or Mooney time, for a $66k hull value. Now, a little over 400 hours, 350 complex/Mooney, and an instrument rating, I just renewed for $1k
Hope this helps--good luck!