The word 蹄 (ひづめ) means hoof in Japanese. Since the meaning is similar to つめ (claw), and since it's spelled specifically with づ, it seems like this word is made up of two parts:
ひ ＋ つめ
I can't figure out what the ひ part might mean, though. It would make sense if it meant a large four-legged animal, or if it meant horse or something else along those lines. But I can't find any information about it in dictionaries or by searching on Google. (Has the origin of ひ in this word been lost?)
I also looked through lists of kanji by on and kun readings looking for reasonable candidates for ひ, but I wasn't able to find anything that seemed appropriate.
Is it possible it comes from a shortening of 扁爪 (ひらづめ)？