Would it be weird to say "of everything" instead of "in everything"?
Yes, unfortunately, only "in" here be used here, not "of." The full expression is to "see the bright side in everything," and this sentence has included this expression. The expression involves the action "to see [x] in [something]," so "of" is not appropriate here. Ex: Mary is always positive. She sees the bright side in everything. Ex: I don't know what she sees in him. (I don't know what she likes about him.)