Onion and Garlic are prohibited to be offered as Prasadam to God. And are even classified as rajasic/tamasic according to Ayurveda. However Brinjal and Hing are also rajasic/sattvik(according to ayurveda) but they are allowed to offer as Prasadam. Why so?
There are various opinions on which foods are allowed/disallowed. The most famous one is obviously the Bhagavad Gita. The Puranas and Dharma texts have more details. Ayurveda is another such text.
For Srivaishnavas (followers of Ramanuja Sampradaya) the rules of the Ahara-Niyamam text of Vedanta Desika form the authority in regards to what can and cannot be consumed.
As far as asafoetida is concerned, orthodox Srivaishnavas do not consume it however it is is culturally ubiquitous and even many temples use the ingredient.
Brinjal is not prohibited as far as I know in Srivaishnava practice. The Ahara Niyaman explicitly prohibits white Brinjal (some interpret this to include green Brinjal as well) with the implication that the common purple one is allowed. In Srivaishnava practice, Brinjal is not offered to the pitrus, hence not cooked on amavasya, yugaadi, shraadhas etc.
You can read Vedanta Desika's Ahara Niyamam here. Note that this translation has the following for the eleventh verse:
 Do not eat,
- ghee that is served after you have begun eating
- foods that have been cooked twice (reheated?)
- foods served by bare hand or spatulas made of iron, uncooked foods may be served by bare hand
- foods already bitten into,
- old, and spoilt foods
- foods prepared in someone else's house (no potluck)
- foods that someone else wanted or rejected
- foods with smell of earth, or separated by finger nails, salt that you can see
- separated for one self and guests
- sap coming out of trees, except asafoetida.
The original Tamil text doesn't mention the "except asafoetida" part. It is a clarification from the translator which is unacceptable.