LotR is quite popular with my wife and I. It's got quite a few sets out now, so it should take you a while to get through them all before catching up to the latest stuff. Arkham plays similarly, but does have a number key differences. It's also new and hasn't had time to come out with a lot of stuff.
One thing for both though, is to have full deck building options, you'll probably want 2 core sets. You don't need to get 2 core sets, but if you want more options when it comes to making your initial decks, it'll help. If it's one gripe I have with Fantasy Flights LCG lines is that the core sets do not come with all the cards you need to build the exact deck you want. However, you don't have that problem with any of the expansions. It is the one thing I really like about the LCGs, I buy the expansion, and it has everything. No buying a booster box and still not having a complete set to play with.
When playing solo, you might also try out playing two handed. Basically make two decks and play both as if you were two different people.
I'll also recommend you try checking out Sentinels of the Multiverse. Now, it doesn't have any deck construction elements to it. Instead you have different heroes, each with their own deck. You put together a team of heroes and go after the big bad. You can play by yourself, going two, three, four, or five handed, or play with friends. Has quite a few expansions out, so lots of hero combinations.
In a similar vein, you might check out Marvel Legendary. It's a deck builder, but designed so it's you vs the game. And while you can play it competitively against your fellow player, you can also just work on beating the villain together instead. It's got quite a few expansions, so once again, lots of heroes to mix and match, and lots of Villains to fight against.
Next up is Pathfinder Adventures. Pick your hero, build a deck based on what kind of cards can be included and go questing. As you quest, get more powerful versions of the cards and slowly level up the character. Now the decks tend to be super small, expect 12 to 15 cards at the start, and some classes have restrictions on what they can use, so sometimes you've got limited options on what you can go with in your deck. But once again you can play this solo or with friends.
I've also seen Mage Knight: The Board Game recommended to folks looking for something like LotR LCG, but I've not played that. Maybe someone else can chime in.
Keep in mind that Sentinels, Legendary, and Pathfinder are not LCGs. So if you're really looking for a game where you can pour over the cards and look for combos and build a deck just how you want it, then these aren't going to scratch that itch. But they're the closest I can think of that isn't Arkham.
