The Sunnah does not support gift exchange for the occasion of Eid. The only exception is sadaqatul-fitr given to the poor on Eidul-Fitr, and part of the sacrifice given away on Eidul-Adha.
However, if the gift exchange results in a definite maslahah (benefit) for Muslims that cannot be attained otherwise - such as to bring feelings of pleasure and dignity to the Muslim children on these days, or to make da`wah to the non-Muslims or the weak-hearted Muslims, then it may be done. Still, this should not be made a regular practice consistently associated with the Eid. And Allah knows best.
BUT, and this is something else as "gift" in western sense, if it comes to paying zakaat or sadaqatul-fitr, or basicly any kind of charity, it is always better to give it to family-members in need as to others, because the Prophet (peace and blessings of Allaah be upon him) said: “Charity given to a poor person is charity, and charity given to a relative is two things: charity and upholding the ties of kinship.” Narrated by al-Nasaa’i, 2582; classed as saheeh by al-Albaani in Saheeh al-Nasaa’i. In this sense, if you feel that a family member of yours is in need of new clothing for example, a sister or an aunty or your mother, its better to spend the sadaqatul-fitr on a new abaya for them, as on anything else. But in that case it is important to define it as charity instead of giving it as gift.
We can't say giving a gift would be immitating the kuffar, for two reasons: gifting is sunnah (just that it is not bound on occassion), and also, coz for example the habit of gifting on christmas day is not their religious custom, rather a custom which had its origin in different cultures.
Gift giving was common in the roman celebration of Saturnalia, for example an ancient festival which took place in late December and may have influenced Christmas customs. Christmas gift giving was even banned by the catholic church in the Middle Ages due to its suspected pagan origins. So far we can't bound the custom of giving gift on any religious practice of the kuffar, its rather an act of kindness.
There are many hadiths which states that our Prophet used to accept and give gifts. Here a few of this hadiths:
Narrated Abu Huraira (RadhiAllahu 'anhu):
Whenever a meal was brought to Allah's Apostle (salAllahu alayhi wa sallam), he would ask whether it was a gift or Sadaqa (something given in charity). If he was told that it was Sadaqa, he would tell his companions to eat it, but if it was a gift, he would hurry to share it with them.
[Sahih Bukhari - Book 47, Hadith 750]
Hadith no: 767
Narrated / Authority Of: Azra bin Thabit Al-Ansari
When I went to Thumama bin Abdullah, he gave me some perfume and said that Anas would not reject the gifts of perfume. Anas said: The Prophet (SAW) used not to reject the gifts of perfume.
Narrated / Authority Of: Azra bin Thabit Al-Ansari
When I went to Thumama bin Abdullah, he gave me some perfume and said that Anas would not reject the gifts of perfume. Anas said: The Prophet (SAW) used not to reject the gifts of perfume.
Hadith no: 769
Narrated / Authority Of: Aisha
Allah's Messenger used to accept gifts and used to give something in return.
Narrated / Authority Of: Aisha
Allah's Messenger used to accept gifts and used to give something in return.
Hadith no: 773
Narrated / Authority Of: Ibn Abbas
The Prophet (SAW) said, "One who takes back his gift (which he has already given) is like a dog that swallows its vomit."
Narrated / Authority Of: Ibn Abbas
The Prophet (SAW) said, "One who takes back his gift (which he has already given) is like a dog that swallows its vomit."
Narrated / Authority Of: Aisha
I said, "O Allah's Apostle (SAW)! I have two neighbours; which of them should I give a gift to?" The Prophet (SAW) said, "(Give) to the one whose door is nearer to you."
And there are many many more regarding gifting, which states, that giving gifts was a great practice from our Prophet and the sahabas, as an act of kindness. But none of the hadeeths i found bound those gifts on any special occassion.
Farther i didnt find any fatwa, hadeeth or verse from the Quoran which would indicate that our Prophet used to make special gifts for his family members on this special day. But i found many hadeeths describing the sunnah practice of the eidul fitr. InshaAllah i will share them with you before end of ramadan, so that we all can enjoy its benefits together.