I think I have a process worked out that suits the lathe I use.
First, ensure that the quill retaining screw is fairly tight.
Second, loosen the tailstock and push it against the near side of the track - there's a bit of give in it from side to side that can shift the center, and this is the best contact point I've found for centralising the bit. Tighten up again.
Third, use a self-centring chuck and drill in half way from both ends to meet in the middle.
Fourth, never advance the quill more than five or six full turns of the handle during drilling. The further out it gets, the more off-center it gets. Once you get to the end of the fifth rotation, stop the lathe, wind back the quill and move the tailstock up until the bit touches the bottom of the hole, not forgetting to push it against the near side of the track at the same time, then start again. Not only does this mean you get less deviation from central, it also gives a good opportunity to clear the drill bit flutes of debris and let it cool a bit, which is vital when working with resins.
Fifth, when working on thinner holes and especially in resins, use a slightly smaller drill if you can (I'm using a 7.8mm drill for 8mm holes)
Sixth, if working on larger holes, center-drill first using a suitable bit. I don't have a spotting bit, so I use a stepped bit.
I'm trying to mainly use jobbing bits, as I've had problems with brad points, but I don't think that's necessarily important.