How old is your daughter?
An adult rescue cat is usually a great choice if you have children (and in general!), as their personality is already formed and you'll pretty much know what to expect. Also much easier than having kittens around and adult cats tend to be calmer and less likely to destroy your curtains!
Other things to consider...
Do you live in a house with access to a garden? Is it a busy road or a safe network of gardens at the back? Could you fit a catflap?
Are you sure no one in the family is allergic?
I don't really think of cats as a "cheap" pet - mine cost £15 per month insurance, £13.50 vet's health plan (includes flea/worming treatments, annual vaccination, a free check up and 10% discount on everything else at the vet's), they eat good quality food (probably works out to at least £1.50 a day) and get through a lot of litter. Plus other things like Pet Remedy plug-ins for my nervous cat, toys and scratching posts, cat sitter or cattery fees when you go away etc.
And obviously think of your long term plans. Cat's can live up to and over 20 years (average is around 15-18 I think).