One of our customers had problems staying in tune on both of her Gibsons with a tune-o-matic style bridge. Here is what we suggested:
1. Replace the saddles! The sharp metal points of the saddles tend to bite into the strings and don't let them slide freely when the tension changes.
2. Same can happen with the nut slots, especially when the nut is made out of plastic, or the slots are not wide enough. Even though you perfectly tune your guitar, once you strum hard or bend a string, tension can get released and knock your string out of tune.
We recommended installing String Savers- replacement saddles made out of graphite. Graphite is commonly placed in the nut slots to aid string friction, so it only makes sense to put it on your bridge saddles too. They are especially good for people with the notorious corrosive sweat issue, that tends to ruin the hardware on their guitars.
New bone nut is also installed to ensure minimal wear and problems and the guitar is set to go! Very important additional tip: always stretch your strings and tune UP rather than DOWN in pitch. Even if you have some problems with strings sticking, it will help you to stay in tune.