The name pika (archaically spelled pica) is used for any member of the Ochotonidae, a family within the order of lagomorphs, which also includes the Leporidae (rabbits and hares).
Of course, “Pikachu” is a portmanteau of the words, pika, … all » meaning to spark or meaning a sparkle of lightning in the Japanese language, and “ch?”, an onomatopoeia in the Japanese language for the squeak made by a mouse. It seems pure coincidence that Pikachu and the Pika bear so much resemblence - or it may just be a third level of cleverness on the japanese creators’ part.