Note: The open ended questions had so many results that I can't possibly address them all. I will attempt to look for and discuss trends. The raw data is here if you want to look over it.
Results were all over the place on this question, which leads me to believe the number one takeaway is that there are a lot of good systems out there with something to learn from.
Higher focus on roleplaying
By far the strongest trend was that respondents by and large want a higher focus on roleplaying in their games:
"Player driven storytelling."
"Story built into the rules."
"Integration of personality and characterisation into the character creation, instead of just numbers."
"Session 0 with collaborative party backstory building."
"Ways of solving conflicts without resorting to violence as a default."
The perception seems to be that games that have lots of combat rules will tend to focus on combat. By implementing rules about roleplaying, players will focus more on it. This seems to make sense, despite possibly being counterintuitive. Many systems have no rules about roleplaying because they want to give players absolute freedom; yet by implementing hundreds of small rules about combat, games inevitably focus on sword swinging die rolls. Who hasn't seen a group fall prey to this trap?
Help the GM
Our poor GMs. They build worlds, arbitrate rules, and keep the narrative going. It's not an easy job, and the responses show it:
"GM never rolling dice."
"Robust GM tools."
"Rules for GM and specific instructions on how to GM this game."
"GM specific mechanics."
Many of the solutions offered by respondents conflict with each other: some of you want explicit GM rules that cannot be broken, whereas others want to remove dice from the GM, instead offering more tools to quicken the pace of a game. To take a step back from the specific implementations offered, I think it's safe to say that most of us agree there is a lot involved in GMing, and that these solutions come from a place of wanting to make the job simpler.
Every game has something to offer
There were a lot of responses that referenced specific mechanics in games:
"Belief from Burning Wheel."
"One Unique Thing from 13th Age."
"Bennies from Savage World."
"Aspects from FATE."
The last clear trend in the data is a desire for rules simplification.
"Easy to pick up and play, especially for newcomers."
"Character creation and advancement using the same rules to keep everything nice and streamlined."
"Streamlining of Skills."
"Single roll resolutions and Player rolling only."
Final Note: Combat
Lots of responders mentioned specific combat mechanics, but the only trend was a desire for an alternative to the hit point system.