My friend is from Varna, and has a valid Bulgarian ID card but does not own a passport yet. Is it possible to travel to Romania with the ID only? It's only for a small trip.
Yes. EU countries are required to grant admission, along with a right of residence for up to three months, to citizens of other EU countries who have a national ID card. This right may be abridged only in exceptional cases, the most likely being if your friend is found to be a threat to public safety.
There is an additional right of residence of more than three months, subject to additional conditions, but this appears to be beyond the scope of your question, and it is generally off topic for this site.
The relevant EU legislation is directive 2004/38/EC.