All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
The majority of the jurists expressed their opinion that it is desirable to bury the hair, fingernails and blood that one removes from his body. Mayl Bint Mishrah Al-Ashra'iyyah, may Allaah have mercy on her, reported that she saw her father cutting his nails and burying them, and he said: "I saw the Prophet doing so." [Al-Khalaal] Moreover, Ahmad said that Ibn ‘Umar used to do so. Ibn Hajar from the Shaafi'ee school said: 'The scholars of our school are of the view that it is desirable to bury the hair and nails as they are part of the human being. This is reported from Ibn ‘Umar and this is what the four schools agreed upon.'
As regards the umbilical cord, it is a part that was cut off from a human being, so it should be buried, as it is like the blood, hair and nails, and it is like burying the clot or the little lump of flesh due to a miscarriage.
It is desirable to give in charity the equivalent of the weight of the hair in silver before burying it. It is reported that the Prophet sacrificed Aqeeqah for Al-Husayn by slaughtering one sheep and told his daughter Faatimah : "Shave his head and give the equivalent of the weight of his hair in silver." [Ahmad]
Allaah Knows best.