Think of the major Jewish male heroes from Tanach: Avraham, Yitzchak, Yaakov, Yosef, Moshe, Aharon, David, Shlomo. All of them would be great people to name your son after, and in fact these are many of the most popular names given today. And not only today; these names have been popular for something like the last thousand years, from "Moshe Maimonides" and "Yosef Karo" on down.
Perhaps surprisingly, though, the trend does not continue with earlier generations. The great rabbis of the geonic period had names like Amram, Hai, and Saadiah. Not only are these names not familiar from Tanach, it is questionable whether they are even in Hebrew. In the Gemara the situation is similar. The predominant names - like Abayei, Zeira, Pappa, and Assi - are foreign sounding and definitely not from Tanach. The Mishna, at least, is full of Hebrew names such as Hillel, Meir, and Gamliel. But here, too, few of the names represent famous Biblical figures. When Rabbi Yishmael's parents named him, couldn't they have chosen one of Yishmael's more illustrious family members instead? When Rabbi Akiva was born, why didn't they move the yud and take off the alef and just call him Yaakov? (Few female names are recorded from these time periods, so it's hard to tell if they were similarly unorthodox.)
It's interesting to see that the pattern continues even within Tanach. But first, we need to clear up the misconception that every Biblical baby had a name invented for him at birth. In fact, many biblical names are used over and over again. According to a concordance I looked at, there are 10 Michaels in Tanach (not counting the angel), 12 Azaryahus, 4 Yirmiyahus, 3 Ezras, and 14 people named Shallum. There are 8 people named Shlomit, and what's more, 6 of them are male. Repetition of names seems to be the rule and not the exception. It is thus quite striking that all of the really famous names are "exceptions".
So in summary: throughout all of Jewish history until, oh, about the time of Rashi, the names of the most famous Biblical figures were not used. But soon afterwards these names became dominant. Why?
It seems that it was considered presumptuous to use a name which had previously been used by such a great historical figure. To draw a parallel, we name our kids Moshe today, but we probably wouldn't look kindly at someone who actually named their kid "Moshe Rabbeinu". In the olden days, the name Moshe alone had similarly specific connotations. And so for a long time, it wasn't considered acceptable to use that or other similarly famous names.
But at some point in the middle ages, the Europeans came up with a brilliant invention: last names. This invention roughly coincides with a wave of rabbis named Avraham and Moshe, and I suspect it's no coincidence. Once you were referred to by two names, there was no danger that you would be confused with the prophet or king of the same first name. And thus a series of Biblical names with very positive connotations became available for general use. If this has led to the demise of such traditional Jewish names as Rafram and Hai, well, that's just the price we have to pay for progress.