The prophet and his sahabah never gave such thing as an Eidee. This is something that came from traditions through generations.
But that doesn't mean that it is bad, and Islam supports such practices. Here are some quotes:
The prophet (saws) said:
exchange gifts, as that will increase love between you.
إن من أحب الأعمال إلى الله إدخال السرور على قلب المؤمن, وأن يفرِّج عنه غمًا, أو يقضي عنه دينًا, أو يطعمه من جوع
From the most loved deeds to Allah is to bring happiness to the heart of the believer, and to relieve his grief, or to pay his debt or to satisfy his hunger.
It is a form of bringing happiness to children at Eid day.