Here's a unicorn for you. The website d20pfsrd.com is very useful for reference. Use the search bar in the upper right to find what you're looking for.
Your daughter's idea of a Unicorn may not match the game's definition. So I'd just make it a pony with a horn. That can talk.
Better yet, if it is indeed for a familiar, there are stats for a Goat familiar. If you tweak up the mental stats and change the racial skill bonuses and the familiar bonus for the master, that would be a decent familiar unicorn.
Familiars really aren't designed to get big (such as to be ridable) and fight, though, so if she wants to do that you might consider using a variant of the rules for an animal companion* or have her get a unicorn as a cohort. Normally there should be some cost to these in feats or class features, and it wouldn't be available until a higher level, but for the fun of a kid those rules can be bent. If you want some help with specifics, I'm sure I or someone else could help with the numbers.
*The Nature Spirit for Shamans has this ability, though it's not normally available until level 16:
Companion Animal (Su): The shaman's spirit animal (i.e., Familiar) takes the form of an animal companion of her choice, using her shaman level as her effective druid level. The animal retains all the special abilities and the Intelligence score of the spirit animal, but also has the statistics and abilities of an animal companion. If the animal is dismissed, is lost, or dies, it can be replaced in the same way as a normal spirit animal.
A unicorn isn't normally possible as an animal companion, but you'd could get a smart horse and then give it a gore attack and some spell-like abilities.
