In the Varaha Purana, it is stated:
ashtottara-sataih purva pancasadbhis tu madhyama | kanyasi
panca-vimsatya parimanam vidhiyate ||73||
“One round of Mantra Yoga should be either one hundred and eight
(108)times which is the best, or fifty (50)times which is medium or
twenty (20) times which is the lowest.”
According to Muktika Upaniṣad, there are 108 Upaniṣads. Among these are: (1) Īśa, (2) Kena, (3) Kaṭha, (4) Praśna, (5) Muṇḍaka, (6) Māṇḍūkya, (7) Tittiri, (8) Aitareya, (9) Chāndogya, (10) Bṛhad-āraṇyaka, (11) Brahma, (12) Kaivalya, (13) Jāvāla, (14) Śvetāśva, (15) Hansa, (16) Arunih, (17) Garbha, (18) Narayana, etc. These 108 Upanisads contain all knowledge about the Absolute Truth.
Sometimes people inquire about the meaning of these 108 prayer beads, but because we think there are 108 Upanisads which contain full knowledge of the Absolute Truth, therefore 108 beads are accepted.
Sometimes on the other hand, the Vaiṣṇava transcendentalists (Rasikas) think that the 108 beads represent the 108 companions of Lord Kṛṣṇa in His rāsa dance, and therefore 108 beads are accepted.
Sources:
https://lovenitai.wordpress.com/2015/08/03/3/
https://www.linkedin.com/pulse/japa-techniques-from-hari-bhakti-vilasa-radhika-gopinatha-dasa
https://prabhupadabooks.com/tlc/24?d=1&f=243921