As far as compatibility, that one has been answered - yes.
As far as old 105 vs new Tiagra, new Tiagra is probably better but psychologically it feels weird.
But the real question is what other work/tune up has been done? Sure you can replace the RD but what if it still shifts poorly? Now you've got re-do all the cables (my suggestion) anyway but you could have done that part first before deciding that a new RD was needed. If it were me, I would go for brand new cables and housing, clean up my FD and RD by taking them off, rinsing or ultrasonic clean them, and then put things together. With fresh cables and after a thorough clean it should be obvious if the parts are working as expected or if they need to be replaced. If you're lazy about the cleaning part, or if it's simply cheaper to get new RD, then I understand.