Far as I'm aware, an adult heir is best, though I can't remember if it's mature or young adult which gets the best score.

Page 56 in the manual gives a list of what gets you a higher score. Not sure if they do it in descending order of importance but it is:

1. Your honour - I think unsurpassed is the highest level
2. Generalship
3. Army size
4. Province control - owning more of Japan gets a higher score, but is harder to get because of rebellions
5. Rival army size
6. Your dynasty - an heir, basically

You can only get the best score (and thus longest shogunate) on the highest difficulty level, No-Dachi.

hector13 nailed it, but I wanted add a little.

The difficulty you play on modifies the score. On Wakizashi mode, 3000 points, or a 100 year dynasty, is pretty much the maximum. You have to move on to Katana or higher to be eligible for a 200 year dynasty. I'm not sure what the max dynasty is on No-dachi mode, because playing that difficulty makes me want to bash my head into a wall.

Swordfighting counts for nothing, as far as I can tell.

Max army size is just over 12k (something like 12800?).

Not sure if child or adult heir makes a difference.

I think hector13 is right about "unsurpassed" being the highest honor. Your honor bar might be maxed out, but don't quit until you see that word.

Getting max generalship is tricky, since you can't train the last couple points. You have to battle. If you're quite strong, most people will not want to fight you. That's why I encourage at least one rival to have a big army.

I investigated this some time ago because I wanted to beat the score at the top of the hall of fame (or "Scroll of Honor" as it turns out)

The scoring assets run on a scale of 0-128, or 80 in hex. Basically, everything counts, and if I remember right, i couldn't actually get a 300-year shogunate without modifying at least 1 thing to be above 128.

All of these count:
80 (128) Kenjutsu - practice swordsmanship and win sword encounters
12,800 troops - max out land and honor to recruit this size of army
80 (128) land size - "Vast fief", you know it is maxed if you manage to get the above number of soldiers
80 (128) honor - "unsurpassed", I don't know why this is so hard to get. Perhaps you get a random number of honor points from winning an encounter, and your last gain has to hit exactly 80, not exceed it... just a theory...
80 (128) generalship - just like Madoverlord said. Tricky. All I can add is that sometimes provinces/daimyos revolt, and you get some last, huge, battles.

Heir - I know you need one, but this point can be considered blurry. Is it better for him to be a youth/young/mature adult? What about his dad's age?