If you don't have your heart set on the Jura the Saleve has some lovely family friendly walks, and you can drive/get the bus to the cable car up.

If you definitely want to be up in the Jura you could go drive up to St Cergue and there is a gentle walk (i can't remember the name of it but if i can find it i'll re-post) which takes you round the site of some old ruins (the ruins are disappointing, the walk is nice) it starts from the edge of the village and goes round the trees and up to a really nice look out point - this should be fine for a family and you get a spectaular view of the lake and the Alps.

http://www.wanderland.ch/fr/suisse-rando.html  this is a good website i use as well to find walks, you can serach by area or path and it will give full trails (i.e. ones that take a couple of weeks to hike) but breaks down the local sections and describes the dificulty of the walk.

Alternatively call up the local tourist info, they are generally really helpful

http://www.geotrails.ch/tourismvd/detail.php?PHPSESSID=1f84f3f09afda0f1c61208232ff72da3&trailid=102293

http://www.geotrails.ch/tourismvd/detail.php?PHPSESSID=1f84f3f09afda0f1c61208232ff72da3&trailid=102289

There are some lovely walks out of St Cergue. 

http://www.st-cergue-tourisme.ch/en/themes/hiking-1249-0

It's also very pretty up by Vallorbe, though the nicer walks we've done are a little lower, by the river through the forest. The caves are cool too.

http://www.vallorbe-tourisme.ch/en/index.cfm

If you drive up to col de la faucille, it's quite short to walk up the last bit, up to the ridge, from where you can walk forever.

You can also stop before the top, and there's a very nice walk, parralell to the slope (i.e. almost flat) with nice view on the Leman. It starts in one of the right-hand hair-pin bends. Check out the Gex tourist office, they have a list of walks.

You can also drive up to the Tiocan (above Thoiry, the departure to get to the Reculet). I think there are possibilities for walking parralell to the slope, but you won't be above the tree-line. If you want to drive to above the tree-line, your only option close by is the Saleve.

The other mountain passes on the Jura are similar, St.Cergue/laGivrine has been mentionned, try also the col de Marchairuz, from where you can get up to the ridge.

Thanks for all the replies!  I have a lot to work with here.

List and card we try with our 3 y.o. in Haute-Savoie and close to Geneva :) http://babyloveaupaysdegex.blogspot.fr/2013/08/une-petite-rando-en-famille-fancy.html
Enjoy! :)

I bought a book : balades  - Pays de Gex - Bellegarde (it is in French) - it is wonderful..  I did one Saturday which started in Divonne , and went over Mussy to Mourez and back above the Grilly road  -  this has the best views ever , 340 degrees over the lake , the alps, montreu, geneva annecy and the jura - Stunning.   I highly recommend the book. Walks range from 1 hour to 6 hours (lostly around 2-3)  plus cycle routes . I have done 3 walks and all were beautiful  - i shall do them all  

 I bought it in a book shop in Ferney but i have also seen it for sale in the Tourist office in Divonne 

Sonia, thanks for that link!  Just to make it more explicit, this is it:

http://www.randos-montblanc.com/pages/itineraire.html

It looks like there's lot of well-researched options here, from super easy to much more challenging, and very close by, too!