It starts at the top with owner Bob Kraft after learning the ways of the NFL, he finally found a coach in Bill Belichick and having him find the personnel to help him succeed. Belichick has the final say, but he surrounded himself to with people who know what type of players he wants to play his way. Not something the Dolphins can say in that time period. The Dolphins have had ownership change and owner Stephen Ross seems to becoming into his own as an owner, but the Dolphins have had 9 different head coaches during that time and 6 general managers.
Belichick has surrounded himself with quality coaches and lots have coaches have moved on to other head coaching positions, but Belichick always find a quality football coach to take over whatever the title and not miss a beat. The Dolphins because of the constant change have had too many different assistant coaches. The Dolphins need to find a coach and give him a chance with his staff rather than change coaches every 3 years that's not good for anyone. The coach bring in his players and then when the coach goes after 3 years the next coach is going to want to bring in his own players and not be stuck with players who probably don't fit in.
Belichick and his personnel staff make sound decisions with their roster even if it's not the most popular move. There is also luck involved such as drafting quarterback Tom Brady in the 6th round of the draft and his development becoming arguably the greatest quarterback in history when all said and done. Once you find a quarterback the coach trusts its up putting the rest of the roster together which can be tough with the salary cap. Belichick typically doesn't pay his players the premium salaries of other players and at times he will trade players when their deals are coming up for the best value possible, such as the trade he made when he traded back up quarterback Jimmy Garoppolo for a 2nd round pick because he knew he wouldn't be able to resign him and why not get something in return instead of getting nothing. He's also traded players when they are about to enter the twilight of their careers like Drew Bledsoe and Richard Seymour each player was traded for a 1st round pick. Belichick will also cut players like Lawyer Milloy to save money against the cap.
Belichick sells high, trading players for the highest value, and buy low, acquiring players at minimal to low cost. He got Randy Moss in a trade for only a 4th round pick and he ended up setting an NFL record with 23 touchdown receptions. He does the same thing with trading draft picks. Sometimes he trades down to acquire extra picks or use those extra picks to get other players. Not every acquisition is a hit. This year the Patriots traded a 2nd round pick for Kony Ealy and he didn't make the final roster. He also gave a big free agent contract to Roosevelt Colvin and he ended up being a bust. It shows the Patriots do make mistakes, but for the most part they don't.
The Dolphins could try to take some steps like the Patriots. Sticking to a coach with a front office and have them put their plan in place instead of scrapping it every 2-3 years. The Dolphins could also take longer looks into how they spend money on their roster. Over the years this team has wasted money on players like Mike Wallace, Dannell Ellerbe, Phillip Wheeler, Jake Grove, Justin Smiley, Gibril Wilson, and more and it has hurt their team on the field and in future years because they have money tied up on players not living up to their contracts. The Dolphins could take a more conservative approach on paying players in free agency. They also could look into not overpaying their own players and look to try to get the best value in return. Jarvis Landry is a free agent this year and the Dolphins should consider putting the franchise tag on him and try to trade him. Or they could look to unload a talented player like DeVante Parker and see what they could get in return because he has been a disappointment and his contract is coming up after next year and if he has a big year wouldn't it be fitting it be in a contract year.
Trying to do things like the Patriots is easier said than done lots of things have to go right, but the Dolphins have been trying things that aren't working and changing their front office and head coaches way too many times. The problem with sports in general today is teams don't give their management or coaches the time to try program working. Instead they are looking for the quick fix, but sometimes isn't it best to have patience and faith in the people you hire through the good times and bad rather than jumping ship when things don't go right off the bat.