Canto 1 - Boyhood
Bāla-kāṇḍa
Chapter 29: The History of Siddhāśrama
Text 1.29.3

एष पूर्वाश्रमो राम वामनस्य महात्मनः।
सिद्धाश्रम इति ख्यातः सिद्धो ह्यत्र महातपाः॥

eṣa pūrvāśramo rāma vāmanasya mahātmanaḥ
siddhāśrama
iti khyātaḥ siddho hy atra mahā-tapāḥ

eṣaḥ = this; pūrva-aśramaḥ = formerly the āśrama; rāma = Rāma; vāmanasya = Vāmana; mahā-ātmanaḥ = of the great soul; siddhāśramaḥ iti = as Siddhāśrama; khyātaḥ = it is famous; siddhaḥ hi = attained the results of those austerities; atra = here; mahā-tapāḥ = for Kaśyapa’s son who performed great austerities.

Rāma, this was formerly the āśrama of the great soul, Vāmana. It is famous as Siddhāśrama, for Kaśyapa’s son who performed great austerities attained the results of those austerities here.

“This was formerly the āśrama of the great soul, Vāmana” means “This was formerly the āśrama of the great soul who desired to incarnate in the form of Vāmana.”

Previously, Lord Viṣṇu had executed various austerities here and then He appeared in the form of Vāmana. Therefore, this āśrama of Lord Viṣṇu is extremely exalted. He had executed austerities in order to propagate these austerities, just as He does in Badarikāśrama in the form of Nārāyaṇa Ṛṣi.

The second half of this verse explains the etymological derivation of the name Siddhāśrama.