The phrase "by interest" refers to something that is of personal importance or relevance to someone. There are several synonyms that can be used to convey this meaning, including "by preference", "by choice", "by inclination", and "by liking". Each of these words highlights the idea that the decision or action is made based on personal feelings or desires. For example, one might say "I choose this option by preference" or "I am inclined to go with this idea by liking". Using synonyms for "by interest" can help to add variety and nuance to your language, while still conveying the same basic meaning.