Can someone give a part of his zakat to a family member (not a parent), say an uncle for example specially if he is in need?
The prophet not only allowed charity to be given to family members, but said that double the reward is issued once for helping a family member and one for zakat:
Volume 2, Book 24, Number 545 :
Narrated by 'Amr bin Al-Harith
Zainab, the wife of 'Abdullah said, "I was in the Mosque and saw the Prophet (p.b.u.h) saying, 'O women ! Give alms even from your ornaments.' " Zainab used to provide for 'Abdullah and those orphans who were under her protection. So she said to 'Abdullah, "Will you ask Allah's Apostle whether it will be sufficient for me to spend part of the Zakat on you and the orphans who are under my protection?" He replied "Will you yourself ask Allah's Apostle ?" (Zainab added): So I went to the Prophet and I saw there an Ansari woman who was standing at the door (of the Prophet ) with a similar problem as mine. Bilal passed by us and we asked him, 'Ask the Prophet whether it is permissible for me to spend (the Zakat) on my husband and the orphans under my protection.' And we requested Bilal not to inform the Prophet about us. So Bilal went inside and asked the Prophet regarding our problem. The Prophet (p.b.u.h) asked, "Who are those two?" Bilal replied that she was Zainab. The Prophet said, "Which Zainab?" Bilal said, "The wife of 'Adullah (bin Masud)." The Prophet said, "Yes, (it is sufficient for her) and she will receive a double rewards (for that): One for helping relatives, and the other for giving Zakat."
Sahih bukhari Volume 2, Book 24, Number 507 :
Narrated by Abu Huraira
The Prophet (p.b.u.h) said, "The best charity is that which is practiced by a wealthy person. And start giving first to your dependents."
Sahih bukhari Volume 2, Book 24, Number 508 :
Narrated by Hakim bin Hizam
The Prophet said, "The upper hand is better than the lower hand (i.e. he who gives in charity is better than him who takes it). One should start giving first to his dependents. And the best object of charity is that which is given by a wealthy person (from the money which is left after his expenses). And whoever abstains from asking others for some financial help, Allah will give him and save him from asking others, Allah will make him self-sufficient."
Also, there is a story of a man who gave charity to his relatives and cousins and the prophet praised him for doing so.
sahih Bukhari Volume 2, Book 24, Number 540 :
Narrated by Ishaq bin 'Abdullah bin Al Talha
I heard Anas bin Malik saying, "Abu Talha had more property of date-palm trees gardens than any other amongst the Ansar in Medina and the most beloved of them to him was Bairuha garden, and it was in front of the Mosque of the Prophet . Allah's Apostle used to go there and used to drink its nice water." Anas added, "When these verses were revealed:--'By no means shall you Attain righteousness unless You spend (in charity) of that Which you love. ' (3.92) Abu Talha said to Allah's Apostle 'O Allah's Apostle! Allah, the Blessed, the Superior says: By no means shall you attain righteousness, unless you spend (in charity) of that which you love. And no doubt, Bairuha' garden is the most beloved of all my property to me. So I want to give it in charity in Allah's Cause. I expect its reward from Allah. O Allah's Apostle! Spend it where Allah makes you think it feasible.' On that Allah's Apostle said, 'Bravo! It is useful property. I have heard what you have said (O Abu Talha), and I think it would be proper if you gave it to your Kith and kin.' Abu Talha said, I will do so, O Allah's Apostle.' Then Abu Talha distributed that garden amongst his relatives and his cousins."
the prophet specifically requested that you pay for your expenses first, then give to your dependents afterwards, to prevent them from asking from others for charity. If you are able to help your family members and be a dependent for them it will be best than to have them ask a stranger for help when you are capable of giving them alms.