I had a Yamaha yzf r125. Great little bike, good for an indicated 85mph and great fuel consumption. It had quite a high seat though.
Mine was a cg125, as soon as i passed i got myself a SV650, not a bad bike to get you going for the first year or so, or since it will be a girl's bike the gladius might do it.
My first bike was a cagiva mito 125 had it for years then moved to zx6r (carb model) then 748r and so on and so on.... Think a good starter bike is an kawasaki er650 or Suzuki sv 650.