When looking for a good solo, you're going to look for someone with very good sustain, basically he can heal himself/herself easily while being able to harass the other player and get in a good amount of cs.
For this reason, vlad is very good because of the recent patch, he can sustain very well. Also his pool can save himself from a lot of ganks.
Morg is another choice, while she cant really heal herself as much, has passive spell vamp can heal a great deal every once in a while. Her spell shield is also great against ganks which is why she is another popular choice in mid.
Morde is another good choice, but trickier to use. While he can't really heal, he can make a shield for himself to temporarily absorb damage. This means you can really harass the other laner pretty hard while keeping your hp up.