FWIW, my opinion is that it's your vacation -- if something makes you nervous, and you have alternatives (as you do), I don't see why you should do anything that makes you uncomfortable. I'm sure you'll have a wonderful time no matter what you decide!

I've used booking.com extensively for many, many years, with no problems whatsoever. I never found good rates on expedia, and so quit looking some years ago.

Good luck!

Thank you both for your replies. I guess I'm not fully convinced that meeting my daughter at the airport on the day she arrives and connecting directly to Santorini would be that much more efficient that it would be worth the stress for me of making the connection and the extension of an already long day for my daughter (her journey to Athens will be @14 hours as she has to connect through Munich). For me, since I'll already be in Athens,I have to make a trip to the airport whatever day we leave Athens so the only real efficiency is that it saves my daughter a trip back and forth from the Athens airport. On the other end, either way, we still have to go back from the airport when we arrive from the islands and back to the airport when we leave for the home. The whole trip is not particularly efficient since I will be in Athens 3 times and going back and forth to the islands but that's life sometimes - it's how it worked out here.
Anyway, I 'm sorry I didn't fully explain but I don't mind renting a car and driving short distances to get around an island. I just didn't want to do the longer distance driving that would be involved in a trip to the Peloponnese. I picked Milos over Naxos after reading forum posts and guidebooks and partly because I went to Paros, Naxos and Santorini on my first trip to Greece (many many years ago I admit) and I wanted to go someplace I hadn't been. I will plan to rent a car. Right now the plan is to ferry to Milos on the 7 or 7:30am ferry spend 4 nights on Milos and then take a ferry to Santorini on the 10:15 supercat ferry and spend 2 nights there before flying back to Athens on a late flight - 9 or 10pm. If I could move everything one day later, we could fly into Athens late the night before our flight home and stay at an airport hotel before we fly out the next morning and thereby save the extra trip to and from the airport. I will put some time into that today.
I very much appreciate everyone's input about using booking.com. I've found 3 potential places in Santorini: Annio Flats; Altana Traditional Houses in Imerovigli; and Ira Hotel and Spa in Firostefani. I'd appreciate any information anyone has about these places. I chose Santorini knowing that I wanted to treat ourselves to a place with a caldera view and these seem to have rooms that are in the $300-400 range - I hope worth it..
Thank you again to all that have posted.

OK Alice, thanks for fuller explanation; it requires full info to understand a person's choices. Hope you can push everything a day later. If so, you want a hotel location that makes it easy to return to airport without cost or hassle. Sofitel is right across from air terminal but Very pricey. Instead, look at Hotel AVRA in Rafina, with Free shuttle to/from airport any time even the wee hours - IF you book shuttle at same time you book your room. Hotel Avra - Rafina Town It's right on a beach if you arrive early, and if you arrive late with no dinner, handily there's a restaurant on premises.

PS Because yo uhave a short stay in Santorini, I still would recommend Firostephani, because more convenient to Fira via bus or walking, and Oia via bus. Imervogli out a bit farther. Check map to decide.

Thanks travelerjan. I admit that your strong admonition not to worry so much about making connections has helped me to feel ok about moving the trip back one day so that we fly back to Athens the night before our early morning flights home and not worry about the flight being canceled. I will check out Hotel Avra and will mention it to my friend who has a reservation at the Sofitel the night before she flies out. It looks like I will be able to do it this way. And I will probably pick the hotel in Firosephani on your recommendation too. Really appreciate the advice.

While I can understand that advice can be better tailored the more one knows about someone's particular interests and constraints, and much as I commend travelerjan for her excellent and generous input, I don't think anyone should feel that they need to share each and every thing that might be relevant. IMO, alicefe, you don't owe any of us an explanation. JMO.

FWIW, I would NOT want to have a view of the ferry port from a room in Santorini. JMO.

Kja, what was that about? I scrolled through the whole list of posts and didn't find any recommendation about a Santorini view of the ferry port. ???

@ travelerjan: My mistake. I thought the Hotel Avra (mentioned in post # 26) was in Santorini. Thanks for correcting me, and my apologies for the confusion.