The main argument for joining (and then remaining) in the Euro for Greece was to limit inflation.
Before Greece joined the Euro, the inflation was almost always in double digits, often exceeding 20% or event 30%. The reason for this was that the Government was running a huge budget deficit, which caused an over-supply of Drachmas. The expectation of inflation was also a factor causing inflation, since people were expecting the value of the Drachma to fall quickly, so they converted any extra funds into other "strong" currencies (mainly US dollars, Deutschmarks, or Swiss Francs), causing the value to fall further as a positive feedback loop.
The introduction of the Euro was indeed successful in limiting inflation, since now:
- people now stopped converting their money to foreign currencies
- The value of the currency was determined by the entire eurozone economy, so that the local deficit wasn't immediately reflected in higher currency supply and inflation directly (but still did indirectly).
- The rules of the eurozone in theory required all countries to run a deficit of only up to 3%.
Of course as we all now the third point wasn't followed by Greece, which meant that the money supply in the Country was increasing more quickly than in the rest of the eurozone, which meant that the inflation in Greece was still much higher that the rest of the eurozone.
Since the value of the Euro itself was stable, this inflation was mischaracterised as "economic growth". But the reality was that the costs for people and business were still increasing year-on-year (though much slower than before), which meant the economy was becoming less and less competitive, so the balance of payments was very negative. Since the currency was no longer floating, this could only be paid by continually increasing the national debt.
After the fail of Lehman Brother Bank, there was increased scrutiny in the underlying fundamentals for new loans by the internation money markets, so Greece became insolvent. The eurozone then created a so-called "rescue package" that allowed Greece to pay the eurozone banks the money they had lent, by loaning new money to the Greek government. All this money didn't benefit Greece at all, the only purpose was for the big eurozone banks to not go also insolvent due to an imminent Greek default. The Greek people are and will be paying back this money for more than 50 years.
So the argument for joining and staying in the Euro was almost entirely driven by the desire to limit inflation. Whether that is actually a good or a bad thing is debatable. As always there are winners and losers with both high and low inflation.
The people that benefit from high inflation are:
- Workers with high bargaining power (both public and private sector), that could demand pay-rises to match the inflation.
- People with access to the banking system, since they could borrow drachmas and convert them to real estate, and then pay back the loans with currency that had much reduced value. The instability of the currency also made real-estate attractive, since its value tends to follow inflation, so it was a no-lose proposition. As a personal anecdote, when my parents took a loan for their house in the early 80's, they paid about 60% of their income for the monthly loan payment. When they eventually paid back the loan in early 2000's it was less than 5% of their monthly income, despite high interest rates.
- People that had income or capital abroad where able to retain the value of their assets, whereas everybody else was losing money due to inflation, so their relative purchasing power was increasing quickly.
The losers with high inflation are all the rest of the people who couldn't benefit from the above bullet points:
- Workers with low bargaining power saw their income be reduced year on year since it didn't follow inflation (Pay cuts are always much tougher to swallow than pay increases below inflation)
- People that could not get a loan saw their rent become higher and higher every year, while their income didn't catch up to the increase.
- People that had money in Drachmas were losing money year on year (This is a huge amount of money, just consider the amount of currency in peoples pockets for an entire country).
Finally, although in the question you mention that being in the eurozone is not synonymous with being in the EU, politically there was the argument to be in the "center" of Europe, being involved in as many EU projects as possible, since EU membership was seen as beneficial both politically and economically.