No, I don't think you can omit 'that' in this context.
When you have the following modifiers in the sentence, use 'that.'
the only
the same
the very
the first
the last
all
only
any
no
every
1) The only thing that remains unchanged is our pursuit of changing for the better.
2) All that glitters is not glod.
3) Any paper that you read will give you the same story.
The relative pronoun is allowed to be omitted within specific conditions. Relative pronouns 'whom/which' serving as the object of a verb can be replaced by 'that.'
4) The man whom/that I saw last night was handsome.
4a) *The man [I saw a man(Object) last night] was handsome.
4b) *The man [I saw WHOM(Object) last night] was handsome.
4c) The man [WHOM(Object) I saw last night] was handsome. (Whom- movement)
4d) The man -- I saw last night was handsome.
5) The dress which/that I am wearing is new.
5a) *The dress [I am wearing a dress(Object)] is new.
5b) *The dress [I am wearing WHICH(Object)] is new.
5c) The dress [WHICH(Object) I am wearing] is new.
5d) The dress -- I am wearing is new.
NewHope, in your example,
6) The only thing [that remains unchanged] is our pursuit of changing for the better.
6a) *The only thing [the only thing(Subject) remains unchanged] is ---------.
6b) The only thing [THAT(Subject) remains unchanged] is --------
If you want to omit 'that', you need to change the adjective clause to a present participle phrase as in tdol's example.
7) The only thing [that remains unchanged] is our pursuit of changing for the better.
7a) The only thing [remaining unchanged] is our pursuit of changing for the better.
Step
1. Omit the subject pronoun 'that'
2. Change the verb to its -ing form
