My answer is confined primarily to prayer within the Western European Christian tradition.

Episodes of silent prayer are attested in the Odyssey and the Iliad, but you're right that the practice was regarded suspiciously in the ancient world. However, Pieter van der Horst makes the interesting point that one of the reasons for this suspicion was the use of silent or near-silent incantations for magic/black magic spells. (The idea there being, of course, that the practitioner would not want others to overhear.)

Philo of Alexandria was one of, if not the, first Jewish writers to pick up on Hannah's silent prayer in 1 Samuel as an example. Christian theologians promote silent prayer nearly from the beginning--Origen and Cyril of Jerusalem both prescribe it (from Cyril, for women at least), albeit with rather different motivations. Cassian and Evagrius Ponticus also take on the question from different angles. Cassian is practical: he wants his monks not to disturb each other while praying, and also worries about the forces of darkness overhearing vocal prayers. Evagrius (and, intriguingly, some Syriac authors) focuses on silence as a constructive aspect of prayer, rather than simply the absence of speech.

This doesn't mean that silent prayer was yet "mainstream," however. Van der Horst also points to evidence from Augustine that suggests silent prayer was still somewhat of an aberration in his time. Monastic Rules present an interesting case. In addition to Cassian's guidelines, the Rule of the Master specifies time for silent prayer and the Rule of Benedict can be interpreted to allow it. Other Rules, however, eliminate this opportunity. That is not to say that early medieval monks never prayed silently; however, a time once set aside for silent prayer was eliminated, suggesting a lingering ambivalent attitude.

To be sure, monastic Rules are very clear on the importance of silence in general. However, early rules separate that general provision from periods dedicated to communal and individual prayer. The attitude definitely develops of living daily life as a prayer in itself (flourishing especially in the late Middle Ages), but this is still separate from the idea of "prayer = petition" that dominated in earlier Christianity.

Yet by the end of the Middle Ages, handbooks of devotional practice include instructions on how to pray silently over crucifixes, images. Since fifteenth-century didactic literature is the least controversial genre in the history of the world (I exaggerate...not by much), silent prayer would definitely a standard practice by then. Joan of Arc even discussed her hours of silent prayer by way of defending herself against heresy and witchcraft charges.

There are two possible ways to link the two time periods, both coming out of monasticism: Mass and reading.

First, in the high and later Middle Ages, monks (male only, now) were basically all ordained priests as well, with a responsibility to say Mass. Although late antique liturgical guidelines encouraged priests to do all their prayers of consecration and petition out loud, for the benefit of the congregation, medieval liturgies build in silent prayer. Thus, the same monks whose job it is to pray all day, are used to silent prayer during Mass.

The second is the development of silent reading. From late antiquity, Augustine's account of his astonishment at Ambrose reading silently is well-known--this could be as radical as praying silently. Paul Saenger has argued that the practice of reading silently was cultivated in early medieval western monasteries and spread by the tenth century. This pertains to the topic because of the major use of reading within medieval monasticism: prayer. Reading was the root of prayerful lectio divina, designed to inspire monks to prayer-meditation-contemplation. For them, silent reading was silent prayer. By the late Middle Ages, reading silently to inspire prayer and meditation seems to have been a common practice even among lay gentry newly able to afford their own books.

As one final note, it is sometimes difficult to tell what sources mean by 'silent' reading. The term prayed in one's heart is common, but silent prayer could mean inaudible/incomprensible murmuring under one's breath, still moving one's lips, etc.

Sources: There is a wealth of secondary scholarship on individual figures from late antiquity through the Middle Ages who implicitly or explicitly discuss silent prayer, from Philo to Boethius to Joan of Arc. A good foundational article is van der Horst, "Silent Prayer in Antiquity" (Numen 41, 1994), but see also the critiques and expansions on his work Brouria Bitten-Ashkelony, "John of Apamea and Silent Prayer in Late Antiquity" (JECS 20, 2012). Paul Saenger, Space Between Words: The Origins of Silent Reading (1997), as well as his earlier Viator article (1982), talks not just about silent reading but how 'silence' came to mean, well, silence and not necessarily murmuring.