Unsure if this works mechanically. I suspect that I am in need of wording guidance.

The goal is that you choose a creature, planeswalker, or player to target and Homing Blasters can always tap to deal damage to that target with its second ability. I don't know how to make this name the target when it enters the battlefield. The third ability should add additional targets to the damage, the original and any additional targets are not removed.

I want it go like this.
1. Homing Blasters enters, the player selects opponent 1.
2. The player uses Homing Blaster's first ability to put a charge counter on Homing Blasters that turn.
3. During the player's next turn, they tap Homing Blasters to deal 1 damage to the opponent.
4. The player's following turn, they use Homing Blaster's first ability to put another charge counter on Homing Blasters.
5. The turn after that the player acitvates Homing Blasters' third ability to name Elvish Mystic as an additional target.
6. The player untaps Homing Blasters with Voltaic Key. The player activates Homing Blasters' second ability, and opponent 1 and Elvish Mystic are both dealt 2 damage.

I understand the intention, but only objects in the stack have targets, and the current wording makes some interactions unclear. (Static abilities do not use the stack and so cannot have targets.)

Another thing unclear is whether or not it is supposed to allow the same target chosen multiple times.

One alternative is something like "When ~ enters the battlefield, put a blast counter on target damageable." However, then multiples (even controlled by different players) will share targets, which is presumably unintentional.

There are other alternatives too, but the simplest is probably to use linked abilities: "When ~ enters the battlefield, target damageable. [...] , : Target damageable." The ability that causes damage to be dealt will then be linked to the other abilities. (Note that you will need "when" and not "as" for the first ability.)