A huge Thank You to John Beltran of the AutoCAD Software Development Team for this excellent explanation of how to change the speed of the Tool Palette Rollout and Rollup. I have been asked this question several times as some want a slower speed and some want a faster almost instant click rollup and roll out. Please use caution when editing your profiles and be sure to back them up first before experimenting with them.
"The timer settings for rollup, rollout, the new "roll out then rollup" (called "holdopen") are stored in the fixed profile XML file. By default these settings are not specified and the defaults are used. You can add these settings to the file by editing it to provide non-default values for each of these behaviors.
To make the changes first back up the "FixedProfile.aws" file in your support/Profiles directory and then edit it using any text editor. The file will be hard to read because it is not well formatted, but thankfully the changes to make are near the top of the file.
Search for the word " <AcApData> " (should be right at the top), and immediately after this string paste in the following:
<PaletteRollupInfo PaletteRollupDelay="800" PaletteHoldopenDelay="4000"/>
It's OK to insert carriage returns or additional white space in this file, just be sure to not introduce any returns/spaces in the middle of a word (like "PaletteRolloutDelay" or value).
Each of the values is specified in milliseconds. In the sample above the values provided are the same as the defaults (1/10th second rollout delay, 8/10th second rollup delay, 4 second holdopen delay). These changes are global and affect all palettes.
So to change the holdopen delay to 3 seconds use:
<PaletteRollupInfo PaletteRollupDelay="800" PaletteHoldopenDelay="3000"/>
You will need to restart AutoCAD for these changes to take effect."
Nothing yet exposed in a dialog/UI or sysvar but perhaps in the future as an F3 feature.