General:
* Apart
* Separate
* To the side
* Away
* Off to the side
* Out of the way
In the context of speech:
* Interjection
* Aside remark
* Under one's breath
* Whispered comment
* Subvocalized remark
In the context of setting something aside:
* Save
* Reserve
* Store
* Put away
* Set apart
In the context of putting something aside (e.g. worries):
* Forget
* Ignore
* Disregard
* Dismiss
* Suppress
The best synonym will depend on the specific situation and the meaning you want to convey.