Fallout – How exactly do I get the “Dog” renamed to “Dogmeat”

Revised answer-

Dogmeat as a Companion

Pros

• Can tank enemies, thus keeping them off of you for longer periods. Less useful if you are a melee character, but still nice to have a distraction.
• Doesn't count as a companion against the Lone Wolf perk, so you can get several good bonuses that other companions will negate.
• Can be used to 'Search for Items', which can be useful for finding missed and hidden loot.
• The perk 'Attack Dog' gives Dogmeat the ability to pin enemies to the ground, effectively taking them out of the for a short period.

Cons

• You cannot gear him.
• He does not have a Companion Perk (passive) that other Companions can have unlocked.
• He only has melee. Ranged companions have the benefit of attacking much sooner, so they'll do more damage overtime. Depending on your playstyle, this might be less important to you.
• His pathing/ai can be annoying and trap you, or simply get in your way constantly. Your mileage may very of course, but for me he was too often in my way when trying to stealth around. He never got me caught, but he will stand in a doorway like a concrete block.
• Since he does melee, he is 'injured' more often than ranged companions (although melee companions would suffer same issue). While injured, he is of no benefit to you.

Additional Comments

• It is really up to your playstyle on if I'd recommend him. I prefer ranged companions over a tank one, and ones that I can gear up myself. I like dps over tanking, as I feel I can kill things fast.

Original Answer

For pure damage, Dogmeat is at a disadvantage. He cannot be geared like "real" companions. The fact you can fully deck out other humanoids should be the deciding factor for you.

Ranged companions can hit from range as well, so their overall dps will always be greater than a melee based companion, as they have no downtime(or minimal) caused by getting in range. They also won't be injured as often, as you can kill a lot before it reaches either of you.

As Chase points out in the comments, "It's worth nothing that getting a ranged companion to "tank" for you is a bit more difficult than a melee companion. Reason being, Dogmeat will be in their face most of the time so you simply have to have Dogmeat remain between you and the enemy and the enemy will often times shift focus to Dogmeat. As you can imagine, it's a longer run for the positioning with a ranged companion"

As Yorik mentions in the comment, Dogmeat can be used to 'search for items' which can find loot you've overlooked, or possibly even wasn't accessible without telling him to look. For PC, press E and then choose 'Talk' option (▲) and from there choose the search option, which I believe is ◄. He'll sniff a bit and possibly run off toward loot if there is any near.

The only other quirk to the argument, is that as of right now, Dogmeat doesn't count as a companion for the 'Lone Wolf' perk. Which can add a lot of benefits to your character once maxed. I assume this is an oversight and will be patch, but as of this message it is still true.

It's under Decorations > Misc > Doghouse for x3 Wood, x1 Steel

Your dog(s) will hang around one in a settlement you assign them to. Note that if there are several doghouses within the settlement border, dogs can choose randomly.