"Since" vs "ever since"
It might add the subtle extra nuance of meaning that since you were born he's never not been in the family.
There two ways in which it changes the sentence:
It adds emphasis to the timing of the events in your sentence.
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It resolves ambiguity in the timing of events. Consider the sentence "She's walked the dog since she ate breakfast." That could reasonably mean:
She's walked the dog because she ate breakfast.
She's walked the dog sometime after she ate breakfast.
She has been walking the dog from the time she ate breakfast until now.
Using "ever since" would show #3 to be the intended statement.
Your example sentence is not quite as easy to misinterpret, but it is still susceptible to the same effects.
He'd been in the family because I was born.
He'd been in the family at some point in time after I was born.
He'd been in the family from the time I was born until now.
Fewer words are better, so can't say I'm a fan of "ever since", but I'd rather use it than be misunderstood.
Also, Oddness of sentence containing "since" makes some good comments about "punctual" and "non-punctual" events.
It doesn't change the meaning, but it does add emphasis like an exclamation point would. Like saying "way back when" instead of "back when".
I think that it most importantly resolves the ambiguity of potential implied causality -- eg, is your birth the reason that your family had a dog?
Perhaps your parents discussed plans ahead of time, say, while dating: "Let's hold off on getting a dog until we have a kid."
Tufy was our Chihuahua. He'd been in the family since I was born.
To someone entertaining this notion as a possibility, this sentence takes on a whole different meaning, doesn't it?
Even while entertaining such a notion, using 'ever since' maintains the distinction (at least by current vernacular's standards circa 2014) that the dog was around since before you were born.
Tufy was our Chihuahua. He'd been in the family ever since I was born.