I'm curious as to what kind of mechanics you would like to see in place Crazy Jerome with the kind of change you're talking about.
I'm more interested in awareness of the problem than specific changes. In particular, I'm interested in not assuming that some possiblities won't work because of the way the math has been done in the past would make them not work. A completely new version is the only time you get to productively question such assumptions, after all.
So I think normally that if 20 such potential areas were examined, it might be that only a few were worth changing in this manner, because some option will work better getting away from zero. On the other hand, with 5E looking to create the feel of multiple editions, it will need a lot more options. The more options you have, the more likely a good one will require some kind of ratio effect (directly or indirectly) that makes numbers very close to zero problematic.
With things like damage expressions, you do want to consider as many options as possible. It could be (making up numbers here) that getting rid of d4s in damage expressions has 4 positive benefits and 2 negative ones, while increasing the base damage from +0 to +4 has 3 positive benefits and 3 negative ones. Or it could be the other way around. And maybe none of those positive benefits are so great as to warrant going against tradition. But you don't know if you don't look, and you don't look if you assume that starting at zero is just inherently better and right.
Edit: I do think that 4E was on the right track getting starting hit points away from zero, but overdid it, while not doing enough on the damage expression front--though having things like at-will wizard attacks getting +Int on damage helped a lot there. There are two different things at play with wizards, and naturally you can solve one of them differently than the other:
1) Letting the base damage expression for wizards start a little higher can mean higher base damage on his staff or dagger, or can mean +Int mod to repeatable spells. Either works.
2) The game making as many options as possible at least situationally viable means that his staff or dagger should at least be worthwhile when trapped in melee, for at least some characters--means the damage expression for weapons needs to change a bit.
If you don't value this second one, then the first option may work better for you. I prefer that both be available so as to maximize the options in play. And you might notice that the way they overdid the starting hit points tended to nullify the increase in damage expression that was provided. How long it takes a character with awful\poor\lacking\average\better\good\excellent options to take out opponents with awful\poor\lacking\average\better\good\excellent defenses (including hit points) is something that ideally the game should support a viable range as wide as possible. When you reach the point where "can't even help", then that option isn't really on the table in play. And you probably will reach that well before you get to, say, awful NPC helper aiding against excellent opponent. A tighter scale of to hit bonuses will help, but not if the damage expressions and hit points aren't supporting that scale.
So I'd definitely want some serious attention paid to hit points and damage, and start both at least a little bit more off zero than something like 3E, if only for insurance. The ratio of how long it takes A to take out B will always be in the game. Make the real possibilities there as wide as possible.
Wizard attacks are only a convenient example of why this is so. If you want to make goblins and orcs and kobolds first level threats that continue to be threats in mass for some time, one of the easiest ways to accomplish that goal, right after slow scaling hit chances, is upping the base damage expression. Again, 4E got half of this right by giving such creatures enough hit points to hang around, but this led to grind for most people. Up the damage expression, they can get by with hit points between the 3E and 4E extremes (and scale slower to boot).
Moreover, you can have "secondary" damage effects with real bite, but still noticably lower than primary effects.