Meer-häs-chen is diminutive of Meerhase. This in some German dictionaries is also Seehase, Colypterus lumpus, an ugly fish I would not like to keep as a pet.
Having scrutinized all dictionaries at our University Library, I found two entries what a Meerhäschen is:
A fairy tale from Eastern Europe, retold by the Grimm brothers; follows a short outline of the story. Bad luck.
Now let us consider the description of the animal: It is pretty, and small enough not to be noticed immediately when climbing the princess to hide under her braids.
So Joerg's suggestion seems quite probale to me. The guinea pig got its German name Meerschweinchen = sea pig because it was imported from overseas. There also is a Meerkatze = sea cat, a small kind of ape very appreciated by children. Those names I immediately associate with a sea-rabbit.
Perhaps our sea-hare should be imagined as an undefined small pretty, fluffy, jolly, and maybe bigeyed pet, immediately attracting princesses. The connection of a hare, or the smaller rabbit, changed in the waters of a magic well, with the watery sea hints to a far place of origin making it valuable too. So every listener may make up a pet of his own like when hearing the tale.
By the way: I would prefer sea rabbit instead of the hare, because it sounds fluffier to me.

Wilfried

P.S. - But there is a rabbit looking member of the guinea pig family: the Mara, or Pampas Hare. This would not fit under the braids of the maid - but the Grimm brothers are known to have changed some aspects of their tales ad usum Delphini. If were a prince changed into a small pet I knew where I should hide: under her wide skirts, by the way having a jolly good look. I am sure the fox would give me this advice, too.