Eevee.
Edit: My editor is currently threatening to restrict my account again, so allow me to answer this seriously.
Most starters are bred to be docile, so they’re often your best bet, particularly if they’re from a certified starter breeder or a pokémon professor. Of these, it’s often said that litten is the softest and gentlest (even though it evolves into something decidedly neither of these things), but there are plenty of people who would vouch for rowlet, popplio, oshawott, and every fire-type starter except charmander (and arguably chimchar when it comes to being calm). Some pikachu may fit this description as well.
Besides that, fairy- and/or normal-type pokémon often include those that are soft, gentle, and calm. Fairy-types in particular, as mischievous as they are, hang their reputation on being exactly this, which is why clefairy is a highly popular pet (besides being cute and reasonably easy for beginner pet keepers to handle. However, the fairy-type has plenty of other candidates to choose from, ranging from swirlix to snubbull. The type even boasts two nursing pokémon in the form of comfey and members of the jigglypuff line.
Meanwhile, normal-types cover a wide range of territory, but this territory also includes chansey and audino (two more soft and gentle nursing pokémon), as well as buneary, bunnelby, minccino, skitty, and lillipup (pokémon that are highly popular to keep as pets, especially for young girls). Of course, don’t discount other normal-type pokémon that are somewhat less popular choices for pets but are nonetheless excellent choices for young children, including bidoof, slakoth, patrat, or whismur.
Granted, every type has potential candidates (psychic has espurr, solosis, and munna; bug has joltik and venonat; etc., etc.), but generally speaking, normal and fairy tend to be the most popular for young girls. My point in saying this, though, is that there’s also a chance that neither of these types interest your daughter; while they’re fairly safe suggestions, it’s also worth it to start from what her favorite type is and work from there.
But of course, if all else fails, there’s also eevee. Cute, gentle, soft, usually calm, and can be practically any type your daughter might love. They’re the perfect pets, really.
Until they eat you out of house and home and destroy your property. —LH
Details! —Bill