RenameLink gadget on Commons causes the page tabs to jump around after it loads. (Note that on Vector, you only see this effect on account without extra privileges. Administrators and interface editors already have options in the "More" menu.)
How it looks like:
(I throttled the download speed here to make the effect more pronounced, but it's also visible on fast connections).The RenameLink gadget would need some support from MediaWiki – e.g. a configuration option to display a "Request move" menu option when the user is not allowed to "Move" a page, which the RenameLink gadget could take over (rather than displaying its own after the page loads, thus avoiding other menu items moving around).