There are ways to include the tykes in some of the games we have already but few are explicitly designed to be played with young children and it seems there is great opportunity to explore the options. What I would like to see:
Board Game. I'd prefer a fixed board but tile elements could be fun for the element of the unknown/suprise.
Charming and thoughtfully sized minis, but with a strong element of adventure. Environmental pieces (outdoors - mountains/trees/caves/castles/ships, etc. indoors - furniture/walls/doors/stairs to help bring the board "to life" in addition to playable characters/monsters/villains, etc.
An adventurous setting/theme with strong fantasy tropes (dungeon/exploration/journey/treasure/obstacles/pitfalls/etc), pitched to young kids without being pandering or overtly cartoonish. There is a balance here that can appeal to both kids and the adults that will WANT to play this with them.
Evocative art and theming, again not pitched too cartoonishly or anime-ish.
Simple rule system and mechanics, with flavors of suspense, surprise, heroism and reward(s). Modular goals for treasure, rescue, escape, stealth, would all have strong appeal for little ones and could make for rousing sessions.
I know that's nebulous. I saw at least one recent board game that seemed to fit the bill conceptually but dropped the ball in terms of complexity. I know this can be done well and have to believe it would fill/create untapped demand.