What does "not having a pair" mean?

From http://blog.peta2.com/2011/05/coming-out-twice.html?c=peta2_enews:

I had been trying to keep my vegan tendencies on the DL for years by that point, but it was getting harder and harder to come up with excuses for why I didn’t want my dad’s scrambled eggs or my mom’s chocolate cake. Looking back on it now, I hate how much I hurt my parents by rejecting their cooking for what can only be described as (and excuse my delicate and ladylike etiquette) not having a pair.

I can't make any sense of it.

Solution 1:

"Not having a pair" is a euphemism referring to not having any testicles. Simply, use of the phrase usually implies that someone is "lacking guts", or not brave.

Solution 2:

Whenever you see that, there is an implied "of balls" (male scrotum) at the end of it. This is kind of crude, so some folks will say it this way and hope you have heard the term enough to fill in the rest.

As Bee said, this is a way of saying that the person isn't acting with the courage a proper adult male should exhibit. The insulting implication is that the person in question is acting like a woman (who would not have said anatomical parts). So in addition to being crude, it is also very sexist. However, it seems to be in some kind of rhetorical sweet spot where women don't mind it all that much. Why, I can't say. As your blog link shows, they even use it themselves, or on women (both in this case). This is admittedly weird as it is a factual statement, but everybody knows what is meant.

Another semi-polite way to say this (in the USA at least) is to say the person lacks cojones (often misspelled/Anglicized as cajones). That is the Spanish word for the same anatomical part. Somehow using a foreign word for it makes it a smidgen less crude.