。
She thought she'd like to replace the old vase with a new one.
Literal
She [topic-は] old vase [object-を] new one [with-と] replace want-to [QUOT-と] thought.
Three structural points stack here. 新しいの uses の as a nominalizer for the adjective — 'a new one,' standing in for 新しい花瓶. The と after 新しいの is the comitative 'with,' the same と used in 友達と ('with a friend') — exchange-type verbs like 取り替える, 交換する, and 比べる take their counterpart with と. Finally, ~たいと思う wraps a desire (~たい) inside a thought clause (~と思う), softening the bald 'want to' into 'thought I would like to' — a common hedge in Japanese, where stating wishes too directly can sound pushy.