I'm not a parent but I reckon it's a good idea, a healthy alternative to the crap served in McDonalds.
Do good food at a good price and you should do well.
Whilst letting women breastfeed in your shop is all well and good, you would not make much profit out of them, ie they come to your place, feed the baby, maybe buy a cup of tea or whatever then leave.
Why not add a sideline to keep them there?
ie do nail-painting [it seems fashionable nowadays for women to have fancy designs painted on their nails, and many of them are willing to pay £10 or £15 to have it done properley], or home made jewellry/crafts etc?
You could do a nail painting course at nightscool/local college or rent an area or a seperate room to someone [this would help pay your rent]
These are just ideas for sidelines, try and think of some yourself.

Not sure where you are, but in the UK you may be entitled to help setting this up from various places, such as the Princes Youth Business Trust, they help young people set up businesses if they are 25 or under, OR 30 and under if they are disadvantaged [health problems etc], there is also the goverment scheme if you are claimig benefit, you are still entitled to a weekly payment for the first year of running your business [this used to be called Enterprise Allowance Scheme, not sure waht its called nowadys].

Be aware of the law ie you and everyone else handling food will need a Basic Food Hygiene Certificate [very easy to pass, anyone that fails the exam should be shot!], also worth doing a basic first aid certificate.

You will also need to keep the Fire Brigade happy, ie provide adequate fire escapes etc.
Local council Enviromental Health and Fire Brigade will both visit and advise you for free.

Provide stuff to keep the kids happy, and want to stay there longer and return, ie a few toys, kids books etc, and you could get a cheap telly and video and play cartoons etc.

[I work in a take away, parents bring their kids in, but sometimes the kids get bored waiting even only 10 to 15 minutes and start crying]
[maybe provide crayons and paper etc as well? or if your good at art, maybe run an art workshop for kids once a week etc? small charge for materials [50p or so?] or free if they buy a meal.

Keep an eye on your overheads ie your premises should be a compromise between busy location and affordable rent [ideal location would probably be right next to your city's bus/train station, but the rent there will be very high!

Shop around for your stock, register with all the Cash n Carrys in your city, BUT be aware that much of the stuff in cash n carrys is cheaper in the supermarket! [check the prices carefully].

Good Luck.

john