Technically speaking Hinduism wasn't as successful as you describe it. It went through enormous variations, and the religion we practice today is almost completely alien to the religion of 1000 AD or even 1000 BCE. I believe Hinduism's greatest strength was its ability to absorb the influences of other religions to become competitive again. When Buddhism and Jainism started to dominate the religious geo-political narrative, the various traditional vedic religions evolved and moved away from their ritualistic practices to become competitive again. The Secular tendencies of the various Indian kings helped, allowing the three major religions of the subcontinent to intermingle extensively.
Another important aspect of Hinduism which made it appealing was its general tolerance. For example, one could worship the Buddha or Mahavira and yet call himself a Hindu. Unlike the Abhrahamic religions, the subcontinental religions were not exclusionist. Some Hindu sects considered Buddha and Mahavira to be Avatars of their respective deities, similarly Vishnu and other Hindu deities became part of the Buddhist and Jain cosmology and mythology.
When these religions came into contact with Islam and the other Abrahamic religion, the upper classes of the religion didn't find it too appealing. One reason is that unlike the Other pagan religions, the upper classes in Hinduism (The Kshatriyas and Brahmanas and the upper vaishyas) are a very very large number of people. Thus while the Abrahamic religions were reasonably appealing to the lower castes and classes, since it offered its message of a simpler hierarchical structure, for the Upper class Hindus the message wasn't so appealing since it would involve them subordinating themselves to a higher authority. In other Pagan religions, since this upper class community was much smaller, the Abrahamic religions did comparatively better at conversions, since the lower classes would convert and the Pagan religion would be absorbed into the Abrahamic religion as the followers of the Pagan religions became smaller and smaller in number (I'm thinking like theDruids here, who were a fairly small number in respect to the rest of the population). In contrast since the Upper castes (The Brahmins in particular) were so vast in number, this mechanic of reducing the upper castes to a small minority didn't work.
A third factor is that the Various Islamic rulers who came to India, and introduced the first Abhrahamic religion, needed skilled people to administer the kingdoms. The Mongol invasions severed the Delhi Sultanate from the Islamic World to some extent, and thus the Indian Sultans were dependent on the local upper castes for their administrative skill. Being in large numbers in the Bureaucracy allowed them to mitigate some of the conversion-minded and more oppressive policies of the Islamic rulers. The Jaziya for example was often not as strictly enforced in India as it was in other Islamic nations.
There are of-course probably more factors, but these are some that i've studied and so am reasonably well versed in. The absence of the long running religious conflict which characterized Europe and the Middle East for close to a 1000 years may also have played a role, but here I'm on a much stickier wicket since I'm basing my conclusions on a much poorer (and therefore susceptible to error) knowledge and fact base. Similarly other factors such as the resilience of the Bhakti traditions in South India also played an important role, but i'll let more people elaborate
