A few possibilities come to mind. Considering the probable age and condition of the bike, it's likely that wear is degrading the performance.

The RD pivots might have developed some play, worsening response to cable inputs.
The housing could be worn or gummy with old grease, making cable return slow
The detent's (clicks) in the shifter might be worn throwing off the shift values.
The chain is worn and now has more side flex, making it resistant to the horizontal shifting force.

These are just for starters. Index systems wear and shift response degrades over time. On a bike you ride all the time you barely notice it and compensate by overshifting slightly or other ways until you decide to replace stuff. There are some things, like new cables and housings that might help, but there's no way an old system will shift like a new one no matter what you do.