To stay longer than expected:
* Remain
* Stay
* Tarry
* Dwell
* Prolong
* Protract
* Persist
* Loiter (implies idleness)
To persist:
* Last
* Endure
* Continue
* Hang on
* Persist
* Remain
To be slow to leave:
* Hesitate
* Delay
* Dally
* Dawdle
* Lag
To remain in the memory:
* Haunt
* Echo
* Persist
* Reside
To be slow to fade:
* Fade
* Diminish
* Subside
* Wane
The best synonym will depend on the specific situation and the nuance you want to convey.