The "e" was pronounced (until it wasn't). There are many different reasons it appeared at the ends of words--including no reason at all. Generally, our spelling system has kept it when it made a difference to the pronunciation of the rest of the word.
Was today's silent "e" ever pronounced in the past?
Yes, it often was.
One piece of evidence is the spelling of Old English words. As ermanen's answer states, Old English spelling didn't have silent letters, so if they spelled a word with an "e", they pronounced it with an "e".1 Thus, words like line2, sun (OE sunne)3, and eye (OE ege or eage)4 were once pronounced with two syllables apiece.
Further evidence that final "e" was pronounced, even into the Middle English period, comes from poetry. Consider the following line from Chaucer's Canterbury Tales:
The tendre croppes, and the yonge sonne
In modern spelling, that's "The tender crops, and the young sun." But this section of poetry is written in iambic pentameter, and so this line ought to have more than eight syllables. The only way for it to scan correctly is to pronounce all the "e"s:
/ × / × / × / × / × /
The ten-dre crop-pes, and the yon-ge son-ne
Was final "e" always pronounced?
In Old English, yes. In Middle English, not always. Take another line from Chaucer:
Ful ofte tyme he hadde the bord bigonne
Here we must pronounce some of the "e"s, but not all of them!
/ × / × / × / × / × /
Ful of-te tyme he hadde the bord bi-gon-ne
The introduction to my edition of Chaucer5 says that final "e" is often elided (i.e. not pronounced) if the next word begins with a vowel or "h". That explains why "tyme" is one syllable. "Hadde" doesn't fit that criterion, and therefore illustrates the other possibility: "Before initial consonants -e is ordinarily sounded, though there are cases on almost every page where it must have been either slurred or entirely apocopated" (my emphasis). In other words, in Middle-English poetry (and probably also in casual speech), you could leave the "e" unpronounced if you wanted to.
Why did they stop pronouncing the "e"?
Besides the convenience of poets, there was a much bigger reason why people wouldn't pronounce the "e": they didn't have to. Language frequently "wears down" over time; pronunciations get more efficient by eliminating sounds. Over time "boatswain" becomes "bo's'n." This reduction happens especially often with unstressed vowels--like final "e", which in the Middle English period was pronounced as a schwa. Why pronounce "name" as "nahm-uh" when you can just say "nahm"?
Note: we do still pronounce some final "e"s--when they are surrounded by certain consonants. For example, consider "cooked" vs. "booted." The "e"s in both words used to be pronounced ("cook-ed" and "boot-ed"). But today, the "e" in "cooked" is unpronounced, as if the word were "cookt," while the "e" in "booted" is still pronounced as a short, unstressed vowel. The "e" in "booted" can't be removed without making the "t" of the stem blend in with the "d" of the ending. It still has a reason to be there, so it gets to stay.
But if the "e"s were unnecessary, why were they there in the first place?
Because they used to carry meaning. In earlier forms of the language, the ending -e formed several different grammatical inflections. For instance, ermanen's quote from the Cambridge Encyclopedia of the English Language mentions that hus is the Old English word for "house," but huse is the form when the word in the dative case. Some other sources of final "e":6
- Some Old English nouns ended in -e even in the nominative case (e.g. ende)
- Some Old English nouns ended in some other vowel, which came to be pronounced as a schwa in Middle English and was therefore respelled as -e (e.g. OE nama -> ME name)
- Some words were borrowed from French, which also hadn't stopped pronouncing its final "e"s yet (e.g. corage "heart")
- Some adjectives took an -e ending in certain grammatical constructions (e.g. the word for "young" is yong, but when it is preceded by the definite article, it becomes yonge: "the yonge sonne"7)
- One way to make an adjective into an adverb was to add -e (e.g. bright -> brighte; this pattern was eventually replaced by the modern adverb suffix -ly)
- Several inflections of verbs had an -e ending (e.g., the first person singular: "he singeth, I singe"8)
And, yes, there are some words that add an -e without any etymological or grammatical basis at all. My Works of Chaucer introduction calls this an "unhistoric" -e. Googling the phrase "unhistoric final e" turned up several books that talk about the phenomenon, but the usual explanation seems to be, "Who knows?" Sometimes language just changes and there isn't an obvious rationale.9
One quasi-explanation is the process of analogy. If most words in a particular category end in -e, then the few words that don't may start adding -e to fit in. One source10 says this happened with grammatically feminine nouns: since many Old English feminine nouns ended in -e (at least in some forms), other feminine nouns with different endings in Old English started adding -e in Middle English. It just seemed like the right ending for a feminine noun to have.11
In any case, the fact is that a lot of words came to have an "e" on the end, and also that most of those "e"s were dropped from the pronunciation.
Some "e"s, before they disappeared, affected the pronunciation of preceding vowels and consonants (e.g. rid vs. ride, rag vs. rage). They were kept in the spelling because they could still indicate those changes.
The new coinage Skype is patterned after words in this category. The "y" needs to be a long vowel to sound like sky, so the word's creators tacked an "e" on the end to make sure it would rhyme with type and not with tip. I, too, have known non-native English speakers who pronounced it as "Skypee." But it is following a standard English spelling pattern, even if it's a somewhat baroque one.
Some "e"s didn't affect pronunciation at all (e.g. ende, blisse). They were kept in the spelling because spelling tends to be conservative, continuing to represent pronunciations that nobody uses anymore ('knight', for example). But eventually, we got rid of most of them.
Stores, shopping centers, and neighborhoods--at least in the U.S.--have lately begun adding these superfluous "e"s to their names right and left, in an effort to seem "olde-fashioned." My grandmother got me into the habit of mocking this trend by pronouncing the "e"s like your students do: "Oldee Townee" and so forth. What can I say? We've got a weird spelling system--we might as well have some fun with it.
1 The pronunciation was probably /e/--more like a Spanish "e" than the modern English "e".
2 "Line," etymonline.com
3 "Sun," etymonline.com
4 "Eye," etymonline.com
5 The Works of Geoffrey Chaucer, New Cambridge Edition, 2nd edition, edited by F. N. Robinson.
7 These distinctions were the remnants of the Old English system of "strong" and "weak" adjective declension. To get a feel for how it worked, check out the similar usage patterns of German adjectives.
8 Pronounced "sing-guh," not "sinj." This ending is the same in Old English and, incidentally, also in modern German: "Ich singe." It's also the source of the "e" in have (OE hæbbe).
9 I find the explanation about printers being paid by the letter particularly unconvincing. This extra "e" phenomenon was widespread in the writings of Chaucer, who died forty years before the printing press was invented.
10 The Prologue to the Canterbury Tales: The Knightes Tale, the Nonnes Prestes Tale, edited by Mark H. Liddell: "Elements of Middle English Grammar: Inflection," p. xxxviii
11 Of course, Liddell goes on in a footnote to mention some masculine and neuter nouns that also added -e, and once again we're back to, "Who knows?"