I know you're shocked to hear this—there are a ton of solo-friendly board game Kickstarter campaigns going on right now! The ones I'm personally planning to spend big on still haven't hit yet, but there is plenty out there that is quite tempting...
1) Escape the Dark Castle
Escape the Dark Castle was a successful KS project last year, and I am glad I backed the first printing and first expansion. It's a fun, old-school romp that has you facing obstacle after obstacle as you attempt to flee the dungeons of a creepy old castle. The art and flavor text are great, and the game is just fun overall. My students love it at game club, and it's also a fun filler on game nights at my FLGS. I think the game is good solo, but I am not so committed to it that I am getting everything plus the collector's box. There are about six days left in the campaign, and if you haven't gone for this one yet, it's definitely worth your consideration. I reviewed the game on this site, and I'll leave my own YouTube video about it here:
2. Solomon Kane
This baby is waaaay out of my price range at the moment, but it's an interesting-looking game with a bunch of miniatures. It's a cooperative game in which the players are virtues who bolster Solomon Kane's faith and prevent the shadows from overtaking him. It's a chapter-based game with a lot of story to play through, and in addition to co-op mode, you can actually play it as a true solo using a special virtue called "Providence." I'm not 100% sure I like some of the thematic stuff going on in this game—there are some "tribal" and "Arab slaver" miniatures that caused me to raise an eyebrow—but I don't know the storyline well enough to make an informed comment.
3. Days of Discovery
This is a quirky-looking little card game that is conceptually interesting. It's set in Portugal in the late 1200s, and it's about planning a journey to discover a new land. Not only that, but it's the first game in a series, which means that it's part of some sort of overarching story. In the first round of Days of Discovery, you must secure a patron, because why travel using your own money? Far better to find a rich person to sponsor you! Next, you'll gather your crew and collect supplies. And in the third and final phase of the game, you will finally sail off into to the distance in search of a new land. The art isn't the best, but the game itself seems interesting, and at $29 (including shipping in the U.S.), it's definitely worth thinking about. Boardgames & Bourbon has a nice video about this one, with comments on both the multiplayer and solo modes.