I've owned more than a few Strats in my time and - in my experience - some of the points in the above thread are accurate, some, well, less so.
Let's take the parts of the instrument one at a time:
Neck - this must be in good order and attached well, no loose screws, etc this only leads to grief. With one Strat I owned the land behind the nut was a tad on the long side before the downward curve to the headstock. This led to the low E and A strings fouling as they passed to the machine head
- took ages to spot!
Body - security is the watchword!
Machine heads - so long as they don't slip, they have no influence on tuning stability. Lockers are simply quicker and easier to change. The real trick is attaching the strings properly and having a minimum of turns around the post to drift. There are lot of "how to" guides on the web, even Fender do one.
Nut - the most common problem. A tight nut means the slightest movement and the instrument doesn't return true (even a finger bend'll do it). Trem down it comes back sharp, trem up it comes back flat - a dead give away. As for roller nuts, the only Strat I've never been able to keep in tune for half an hour's playing at a time (and I'm a Blackmore fan) had an LSR roller on it. Diabolical contraption, totally unpredictable from new. Loose nuts have fewer tuning issues, they manifest in other ways. I use a dab of 3-in-1 on my stock nuts for extra surety(oo-errr).
Pivots - in my experience they rarely give real problems, either twin post or six screws, until well worn, they seem to be pretty much trouble free. After a basic alignment activity on the "classic variety" I've never had to tweak one. With the modern US one I've worn a set of pivots and hence replaced the bridge and posts - this didn't lose tuning but the movement and tune/bend weren't smooth and predictable.
Springs - So long as they are free to move - I've had Strats where they foul either the backplate or the body cavity - only affect the balance point against the strings. This is seen as how far the back of the bridge is from the body, or how much up bend you can pull.
Saddle edges - so long as they are free from sharp (as in cutting) edges all should be fine.
Strings - Apart from fitting them correctly and stretching them in on installation, there is no issue here until they are past their life, when they'll start wandering everywhere, but that's not the Trem's fault. Fender Bullets are no better than any other string type, they merely "go off" quicker and can be harder to take out at change time if they break!
Hope the above helps