The Solomon Islands overall is a safe place to visit, BUT, Honiara can be unsafe and dangerous in certain areas for a foreigner and especially as a solo woman, and there are some considerations you need to keep in mind when travelling to the islands.
Most of Honiara follows the Honiara-Lambi to Honiara-Aola road and Kukum Highway. Pick pocketing is common, and if you have your bag on your back, you are at high risk of getting things stolen. Wear your bag across your shoulder and under your arm, or on your front, so that you're not an easy target.
I did put my bag on my back after buying some vegetables as I figured they were low cost and it was more comfortable this way. I was pleasantly surprised to have three separate women come up to me within the space of 20 minutes and warn me that I'd likely get pickpocketed if I didn't secure my bag. I then put the bag on my front so as not to worry any kind passers-by who were looking out for me.
If you walk away from the Honiara-Lambi / Honiara-Aola road by yourself, you may get into trouble, especially if you're carrying luggage and look like a foreigner. On my first day, I went for a large walk to check out a few accommodation options with my backpack, including away from the main roads. I passed a group of a dozen men drinking in an alley and they followed me, calling out to me and telling me to come with them. I ended up making a run for it into a nearby local accommodation where I waited with their female receptionist for an hour until the men got bored and left. The receptionist was pretty angry at me for walking around the area with my backpack on, and I couldn't blame her for being mad. When they left, we ordered a taxi for me to King Solomon Hotel, the most common hotel for foreigners.
I did walk through this area again later without luggage as I had organised a meeting with someone over this way. I got a lot of catcalling, a few attempts at inappropriate touching, and one man try lift my dress up with a stick (I was wearing shorts underneath anyway). This stuff doesn't bother me too much, but if it makes you uncomfortable, I'd recommend to stick to the main road if you're by yourself (Honiara-Lambi to Honiara-Aola road), and don't walk past China town.
If you're with a male traveller as well, you'll be pretty fine, but it can be a bit stressful as a woman. Sexual harrassment is common, and sexual assault isn't uncommon. But this is just in Honiara - the rest of the Solomons is fantastic and very safe! An easy get-around if you're a solo female traveller is to organise not to spend minimal time in Honiara - just one night at the start and end of your trip!
Keep an eye out online before you go. Whilst protests don't happen often, when they do they can get very violence, include burning of building, looting, and rampant lawfulness. If there's any protests in the news, reschedule your trip. You do not want to be in Honiara during this.
Cyclone season in the Pacific goes from November to April. That does not mean you can't travel during this time, but it does mean you MUST get travel insurance! I've actually only travelled during cyclone season, as most of my Pacific trips have been to do research as well as tourism, and I've found it's easier to secure interviews in cyclone season. But make sure that you have a travel insurance that covers you if a emergency strikes, and then when/if things do go wrong, don't stress! Give your insurance a call, let them know what's happening, and enjoy your extended trip and the bad weather.
I recommend FastCover for the Pacific - I am not sponsored by them, but they've helped me out amazingly in a few difficult situations, and they've been beyond reasonable and incredibly helpful.
This information is same both for your safety, and for being an ethical traveller.
When in rural areas, please dress modestly. Men shouldn't be walking around shirtless, and women should have covered shoulders and skirts/pants/shorts past the knees. It is so important to respect people's culture when you are a guest in their country. This will also make you less of a target for pickpockets. If you're walking around shirtless (men), or wearing a miniskirt (women), you'll stand out as someone who definitely doesn't live in Honiara and make yourself an easy target.
Many villages are catholic or seventh day adventist. In the villages, please wear modest clothing as well. Bikinis are not appropriate, and would be rather shocking to a village that hasn't seen women wearing bikinis before. It's insensitive, and would make you come across as rude and even entitled. Respect the people, respect their culture.
Please ask for permission before taking a photo of anyone. This includes people in the villages, and especially people in traditional dress. Make sure that they are comfortable with you taking their picture, and if they say no, respect that. It helps if you show them the picture you took as well, so they can see what it looks like. In some remote areas, people may not have seen pictures of themselves often and may really enjoy seeing what you've captured!