Why is "meta" pronounced differently to "beta"?

Solution 1:

I guess the easiest explanation is that the vowels that are represented by the English e are different in the Greek words.

beta => βήτα
meta => μετα

Pronunciation of (originally) Greek words and their individual phonemes is, alas, not always straightforward, consistent or logical.Different words were borrowed at different times, and from different sources (straight from Greek, or through Latin, Italian, French, etc.).

In general, we should also make a distinction between words and prefixes from Greek, and the names of Greek letters. Especially in the case of the names of Greek letters, there are many differences in pronunciation, as you can see in the link that Josh61 mentioned in his comment. Why do we pronounce those names in so many ways? Well, it seems that people have been using Greek letters in science for a long time, and the first thing we learn when we learn Greek is reciting the alphabet. However, modern Greek pronounces those letters quite differently from how the ancient Greeks used to do it, and our understanding of how they were pronounced has evolved over the years.

At the same time, the names of the letters are used in everyday speech and they are used by people that do not study classical Greek. Those speakers may well pronounce the names of the letters based on the English spelling.

With words that were borrowed, pronunciation often has little to do with spelling (rather, spelling follows pronunciation!) and the (perceived) pronunciation of the borrowed word is often largely retained.

So when we look at the η, we notice that it usually gets rendered as an i: in English (similar to modern Greek), for example:

Ἀκαδημία -> academia
ᾍδης -> Hades

When we look at the ε, it usually gets rendered as e:

μέλας -> (melas) -> melancholy
μέθοδος -> method

There are, however, exceptions in both cases, like ἐγώ -> ego.

So, in short, beta gets pronounced the way it does because it is not an exception to the general rule that an η becomes an i: and meta gets pronounced because it also is no exception to the general rule that an ε gets pronounced as an e.

However, there are plenty of exception to those "rules", especially when it comes to the pronunciation of the names of Greek letters.