What is the history of silent prayer? Correct me if I'm wrong, but I don't believe there is much in the way of Biblical or early-church precedence for praying silently.  It seems like prayers were generally spoken aloud.  In modern times, especially in evangelical circles, it seems that silent prayer is widely accepted and commonplace.  So my question is, when did it become a normal thing?  Was there some specific person or writing which significantly swayed popular opinion, or did it just happen organically over time?
There is indeed much evidence that silent prayer was taught and practiced in the early church. Several examples from the first few centuries will demonstrate this:
Clement of Alexandria (150–215) writes:

Prayer is, then, to speak more boldly, converse with God. Though whispering, consequently, and not opening the lips, we speak in silence, yet we cry inwardly. For God hears continually all the inward converse. (Stromata, 7.7)

Cyprian (200–258) writes regarding Hannah:

she prayed to God not with clamorous petition, but silently and modestly, within the very recesses of her heart. She spoke with hidden prayer, but with manifest faith. She spoke not with her voice, but with her heart, because she knew that thus God hears; and she effectually obtained what she sought, because she asked it with belief.  (Lord's Prayer, 5)

As far as the prevalence of the practice, this is a bit more difficult to confirm, but we do know that Cyril of Jerusalem (313–386) commanded it of women in his catechetical lectures:

let the married woman also [...] pray; and let her lips move, but her voice be unheard

It's also specifically commanded in the canons of the Council of Laodicea (363):

After the sermons of the Bishops [...] there should then be offered the three prayers of the faithful, the first to be said entirely in silence, the second and third aloud

Similarly, in the Apostolic Constitutions (380), Book 2, Section 7:

After this let the sacrifice follow, the people standing, and praying silently

Summary
It's tough to demonstrate exactly how common the practice was among individual Christians in the early church, given the documentary record that has survived. But the record clearly shows, particularly the last two documents cited above, that silent prayer was widespread in Christian churches by the 4th century, if not earlier.