The abilities I use for everything are Fulfillment, Ingenuity, Preserve, Maker's Muse and Brand of Fire.
Simple and effective set for mashing Standard Synthesis, but far from an ideal ability set for HQing essentially anything.
Granted, it really depends on how devoted to ranking
all crafts you are, as to what abilities you would want to use or have access to.
One of the easiest factors to pin down when HQing an item is "What rank is the synthesis I am trying to HQ?"
If you're R50 and trying to craft a R10 item, you have essentially no use for "Increases success rate of X" and should be hammering on other things such as "Increases quality (decreases success)" or "lessens durability loss". Inspiration (GLD 36, quest) combined with Innovation (R30 ALC, auto) can quadruple the quality gain on a single Bold - easily granting 100+ quality!
On the other end of the spectrum you have R50 synthesis (Or anything 45+ really)... These are where "increased success rate" can come in handy. Don't overlook powerful tools such as Hand of the Gods (R35 ALC, quest) or Perfection (R36 CRP, quest) which can squeeze more quality or progress out of a synth that would otherwise be difficult without them.
Some overlooked skills that can be useful (Not entirely personal experience..)
High Return (R36 BSM, quest) when used later in a difficult synth, it dramatically increases success rate when said rate is very low.
Harmonize (R10 CUL, auto) lowers the risk of a synth, and is one of the few (Maybe only?) skills that can remove Aetherial Sparking. Useful for Finished items to allow more spammage of Touch Up.
But don't mind any of this, or even bother to learn about HQing unless you're a glutton for punishment. Changes coming in both 1.19 and 1.2 makes all of this null.