Hi Jen. A portable gazebo sounds like a great lighting project, wish I had one. I agree the flame type lighting will not work so well here, in particular because of the canopy. If it were me, I'd take another look at solar, but not the all included solar lights you see at the garden or home building centers, they never provide much light or hold up long.
Solar does work, we've been powering our entire home with it for 12 years, washing machines, freezers, TV's and coffeemakers, and lights. In your case you need three things to pull it off nicely. A small panel, along the 10 or 20 watt size, easy to find on ebay or other supply houses, a small battery, and the right lights.
The lights are the easy part, but make sure it's something fun. I'd go with LED module strings around the perimeter, but there are some interesting landscape lights too. The real beauty of these type lights is they generally work on 12 volts DC, and so does the solar panel and battery you'll need. I'll clip a link below to one good source for these type lights. We use the LED strips in our kitchen, den and workshop now, they also run straight off our battery bank, which is charged by the solar array.
For a battery, you need to decide how much storage you want, and how much space and money you're willing to invest. A good small and inexpensive battery is the sport battery, usually 12 volt and 8 amp hours or so. They are the size of a 16 ounce water bottle, and can be found in places that sell fishing gear, hunting supplies, or hobby supplies. If you go with one, make sure it is a sealed gelcel type. Usually they cost less than $20 USD. One of these will run a strip of LED's around your gazebo for a couple nights, all night long. A larger alternative is a lawn tractor battery, $30 or so, the size of a shoe box, or a boat trolling battery, maybe $100, but enough storage for weeks.
The simplest way to get a solar charging system going might be to go to an auto parts store and get a battery maintainer, about the size of a shoe box lid, they are designed to sit on your cars dash and charge the battery with the sun through the cigarette lighter outlet. If it were me, I'd get either a sport battery, or lawn tractor battery, have someone find a place for it, then wire in several cigarette lighter outlets to it, one near the led lights, one near waist level, and one near the canopy. The solar maintainer can plug into the one near the canopy, the LED's into another, with a switch, and then you can plug in a small radio, mp3 speakers, or some other small electronic device for music or entertainment. The maintainer will charge the battery in the daytime, then the battery will run your devices at night.
These types of solar chargers are very small output, usually measured in milliamps. You can plug in more than one if you need to and double up on solar charging, since they have charge controllers built in, they won't overcharge the battery, even if you plug 5 of them in, they will just charge the battery up faster. You can also buy a separate solar panel, but you might need a charge controller, depending on the size of your battery, you can find them on ebay too.
We have several out buildings that are solar powered for lighting and small hand tools using this same type system, they have been working fine for years. Solar really does work, you just need better equipment than the cheap garden lights you see for a few dollars, the solar panels are just too small on those to really do anything. Good luck with your project Jen, and take care, Rudydoo