终于 (zhōngyú) means 'finally, at last' — it emphasizes that something happened after a long wait, effort, or delay. It's the standard, neutral choice. Compare: 总算 (zǒngsuàn) is more colloquial and often conveys relief that a desired outcome came; 到底 (dàodǐ) can also mean 'finally' but has a separate emphatic/question sense ('on earth') that 终于 lacks. 终究 (zhōngjiū) adds a feeling of inevitability.