Hotels close to Ben Gurion (TLV)

First, the busiest online Israel forum is on TripAdvisor, so in addition to any replies you get, here, head there for all your questions: https://www.tripadvisor.com/ShowForum-g293977-i1733-Israel.html

From that forum, I know that there is only one hotel anywhere near the airport, and it's not nice, and has nothing nearby except a hospital (meaning no restaurants), and while it's close to the airport there's no transportation from this hotel to the airport.

So, the easiest thing to do is stay in Jerusalem or Tel Aviv, both of which have multiple transit options to the airport, depending on which day and what time of day you're going (both are important in determining which options will and will not be running).

Note that you have to be at the airport 3 hours before your flight departure, regardless of when it's leaving. It's frequently mentioned that the airport is hopping in the midnight to 6 AM time period (when many visitors assume it will be dead).

As Harold said, just stay in Tel Aviv or Jerusalem. If you need to be at the airport before 6 am, there won't be any traffic on the road. It's about 20-25 minutes by car from Tel Aviv, 40-45 minutes from Jerusalem. You can take a taxi from Tel Aviv, it should be a fixed rate of $35-40 (based on the current $ exchange rate). If you want to save money and don't have a lot of luggage, you could take a taxi to the train station in Tel Aviv for about $10-15 and then take the train to the airport. Trains run every hour all night, but not on Saturday (Shabbat). From Jerusalem a private taxi would be $70-100. Or you can take a shared taxi for $20 per person with pick-up at your hotel. The journey will be a little longer, since the taxi will pick up other passengers as well, so you may have a mini-tour of Jerusalem before getting on the highway.

There are more flights leaving Israel between 5 am and 8 am than any other time of day. That means lines for check-in, security and passport control, so plan to arrive 3 hours ahead of your flight. OTOH Ben-Gurion is pretty efficient and you may be pleasantly surprised to find you have lots of time to relax in the duty-free area.