I agree with Che...IF the strings have been stretched. If they are really new and they haven't been stretched they'll go out of tune again and again. When I restring a guitar I stretch each one by working along it from the highest fret, pressing down the string at that fret and pulling the rest of the string up pretty firmly, then move a fret down and do it again. I sometimes go over it twice that way. After that they stay in tune. Any further problems will then come from a string sticking in a nut groove as Che says or (sometimes on a Strat) catching under a string tree.
"What's that terrible noise?" "Well, it's either the Hoover belt needs replacing or Marc Almond's on Jools again."
I have a HB-35Plus, and it had tuning issue initially too. As @chedapapa mentioned, a little work on the nut string slots fixed the issue. That being said, I followed up that fix by overhauling my HB35. None of it was necessary, but I wanted something different.
As for tuners, the factory ones are OK, not great. Mine did seem to hold tune well, but once I removed them it was clear they were all unique in quality level. So I replaced them, I did make a new nut, but it wasn't necessary, I replace the bridge, but that is only because I added a Bigsby.
As for a roller bridge, you will need to verify the string spacing of the new bridge and the factory HB bridge, they are not all the same. Also, you need to verify the post spacing, as the roller bridge I used had a narrower wider post spacing. Then you need to verify the height (thickness) of the roller bridge. The roller bridge I used was taller than the original HB. This resulted in the bridge needing to be bottomed-out to the body to get the action where I like it. If I want to go lower, I'll have to remove a little material from the bottom of the roller bridge.
I swapped out the pots and switch, and again this was not necessary. The new parts are better quality, but function feels the same.
The only things the original HB35 needed was to prep the nut and dress the frets, everything else could have been left alone. Oh, I replace the original strings which were rough and not pleasant to play.
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