I bought a ticket from Ancona Indiana on my Lufthansa app. Because I bought it well in advance, I paid extra for the Flex fare. My class is Premium Economy.

One of my daughters, who lives in New Jersey, was planning to meet me in Indiana, where my other daughter lives. However, she is now unable to do so.

Since my return flight had a layover in Newark I thought it would be fairly easy to change my return flight to allow a two-day stopover in Newark. Apparently not. The customer service agent told me the delay in departure from Newark had to be less than 24 hours. This would give me about 15 hours there, much of which would be consumed getting to and from the airport. I wasn't happy about this, but I asked him to proceed.

When you change flights using the Flex option, you have to pay the difference in fares. (I knew this.) He said the difference would be over €1000! I checked the cost of a multi-city itinerary: AOI-IND / IND-EWR-AOI. This would cost s few hundred euros more than what I paid for the original round-trip ticket.
​​​​Can anyone answer these questions:
What is this < 24 hours rule? I checked the rules for the Flex fare, and I don't see anything about tha
How do they calculate the price difference?

At the moment I feel that what I paid extra for the Flex fare was a big waste of money.

I've never flown on LH but just in general most every airline I'm aware of have a similar 'under 24 hours' rule for connections. There are a few air travel experts on here and may have specific advice for you. But otherwise maybe ask for advice over on FlyerTalk.

Each airfare class for every airline is a little different. It is important before booking to understand the pros and cons for each ticket. Unfortunately the names of the fares can be somewhat misleading. In your case I believe you can rebook for free but have to pay the difference in the cost of the fares. Sometimes the differences can be quite large especially if the flight change dates are close to your flight dates. I think you can get a refund on your flights for a fee. If the fee is not too high you could refund and book your flights the way you want them.

MJS is correct. Each specific fare "bucket" (not something general like "flex economy," but a published fare with a code that begins with some letter; Lufthansa uses "H" for economy flex in Europe, for example) contains its own rules on things like changes, stopovers, etc.

When non-Lufthansa flights are included in the same PNR, using, say, United for domestic connections within the US, it's possible that there's no exact comparable fare bucket for the UA segments and those operated by LH. The domestic stopover or change rules might be different from those in the LH fare bucket, hence your situation. It's hard to say unless we know the specific fare codes for the flights in question.