Rand and Egwene are hardly what you'd call "close friends".

Rand lived miles away in the woods, and came to Emond's Field only for festivals (that's a few times a year only), and only in recent years he started spending time being embarassed around Egwene (and vice versa) during these rare visits,, because their parents wished them to marry eventually, and they were reaching the age when they're expected to start courting. Rand didn't spend time with Egwene because he wished to, but because of social pressure. Same for Egwene. If she were to marry, it might as well have been with Rand whom she didn't dislike but she was in a phase were she was feeling rebellious about social pressure, and unsure she wanted to spend the life of a married woman. Egwene was deluding herself she might want to be a Wisdom, which would have made her leave Emond's Field and be free - or so she thought at the time, until meeting Moiraine presented another opportunity closer to her desire for independance/freedom. Freedom to do what exactly? Like many teenagers, Egwene didn't quite know yet, except that she had a very strong desire to be her own woman and answer to nobody (which unless she went to live alone far from everyone else, Egwene couldn't accomplish). When the Aes Sedai adventure got her saddled in another way and even more so than in her old life, Egwene started wishing to get her freedom back again, which Siuan's mission let her do. And so Egwene spent a few months "living the life" - restrained only by having Nynaeve along, until Moiraine put the girls back in their "proper place" and Egwene jumped on the opportunity of training with the WO... only to fall under strict authority and guidance again. The rest of Egwene's storyline was about learning what sort of society you get when everyone make what they want of rules, refuse to follow anyone and put their own desires first at all times... by being put nominally in charge of women who did exactly that and had chosen her because she wouldn't have any power to interfer with their agendas. Egwene couldn't even try to do as she was told anymore, she was constantly told to do 10 different contradictory things. And so Egwene began to understand why's there's rules and the role of a leader, with many bumps on the road, such as acting as her old stubborn and mutinous teenager self again and getting captured by the Tower as a result. Egwene isn't "power mad" (the only dimension of power she likes is not having anyone above her, she's actually bored or irritated with being expected to tell others what to do) she is, or was, "freedom mad". One of her "weak spot" will probably always be her tendency to use authority as a mean to garantee herself as much freedom to manoeuver as she can get. She hates feeling bound, and in a context where she has authority, that incarnates itself in dictatorial/absolutist tendencies she needs to keep under restraints (and during her rise, sometimes she failed). All Egwene's early "dictatorial" decisions aimed to restrain the power of others to have a say in her decisions (Egwene hates to compromise and negotiate). Learning that even at the top there are rules she not only must follow but is also responsible as Amyrlin for enforcing, that she always has to negotiate when others are concerned, learning when she can or must use the full extent of her authority, versus when she musn't restrain the freedom of action of her subordinates, is something experience is teaching her.

Egwene isn't a "boyhood friend" of Rand like Mat and Perrin, and in fact Mat and Perrin have known her quite better than Rand (to an extent, there's not that much boy-girl interactions in the TR), both having lived near the Inn for many years.

It's not that Rand and Egwene disliked one another, but aside from encounters under watchful eyes during festivals, they barely knew each other before leaving Emond's Field. In EOTW, you wonder if these two even ever had a genuine conversation, something that was more than telling each other socially expected platitudes while keeping their own feelings hidden. You really get the feeling that when Egwene told Rand she wasn't sure she wanted to ever marry anyone, if was the first time any of these two really told the other something they meant. Not that Rand had the guts to tell Egwene "Well, I'm more inclined by nature to do what's expected of me, but I'm not quite sure I've reached the age when I feeling any desire to court anyone, let alone marry".

As they left, Egwene spent most of her time with Moiraine whom Rand avoided and that amounts to 2-3 weeks (with no sign that Egwene and Rand grew any closer or spent time together on the road) when Rand mostly stayed with boys and men, after which they were separated, reunited very briefly in Caemlyn with no time to get personal, then it was the Blight, and Rand started avoiding Egwene and Nynaeve because of what they had seen. Brief encounters in Fal Dara, and then nothing before Falme, when Egwene was in shock and Rand sick, then isolating himself. They had one personal encounter in Tear, the one at which Egwene told Rand what they both knew: they didn't love one another and wouldn't marry. After that, Egwene was with the WO and Moiraine, and the relationship between Rand and her were mostly bickering and confrontational. Egwene left for Salidar without Rand and her getting any closer.

So by most people's standards, Egwene and Rand are just acquaintances, with the added dimension that a marriage was wished by their parents, forcing them to interact awkwardly on a personal level for a while while not being intimate enough with one another for this. Rand had for Egwene a certain attachment (and vice versa), and the basic feelings they have for people from home, sharing the same socio-cultural background. He also has toward Egwene the same attitude and feelings he was raised to have for women. And the same for her.

So why exactly should Rand include Egwene on his list? She doesn't have, nor ever had, a close relationship with Rand. The most meaningful things she's done was to tell him not to flee to go die alone and away from everyone (more at her horror of that notion more than anything).

For the rest, the little they had was confrontational, preventing a real friendship from developping. They never grew out of their teenager caricature of the male-female relationships, Two River style. They've developped more mature relationships apart from one another, but together they're still at that immature stage. Those two are like water and oil. Egwene's advices to Rand have always pushed Rand to do the exact opposite she sought, as any advice from Rand to Egwene would have.

Moiraine did not have a great relationship with Rand either, but she was more mature and eventually he saw her contribution (not to mention that he sort of idealized her the minute she "died", especially that Moiraine's contribution is more "professional" than personal, quite relevant when it comes to his role as Dragon. In the context of Rand's comment (when he's listing people who helped him remain Rand the man), it's still a bit odd that he would include Moiraine.

Tam of course is by a long shot the biggest influence on Rand's character and values.

Elayne in all fairness doesn't fully belong on the list much more than Egwene, but she's had a few meaningful episodes with Rand of an emotional nature that count, not to mention her feelings for him through the bond which played a major role during his epiphany - and from Rand's perspective, she most definitely belongs there.

Min does fully belong there of course - she's the one among his three lovers who had the most personal influence over him. Aviendha belongs there too.

Mat and Perrin's parts were rather passive, especially Mat's. He's the one who adjusted the worst to the fact Rand became the Dragon. If Rand includes him in there, it's not so much because he's been there as a friend since the end of TEOTW, but more because despite distancing himself from Rand on a personal level, Mat has always remained as loyal to Rand as he used to be while they were closer friends, and in the context of Rand's thoughts, that counts for a lot (especially when you compare his situation to that of LTT, who had the likes of Demandred as lieutenants).

Nynaeve's inclusion doesn't even need explanation.

Who's missing from the list isn't Egwene, it's Cadsuane, a reluctant inclusion of Cadsuane as, even though Rand doesn't like her or her methods, post-Epiphany Rand is fully aware of how instrumental she was. It's her who made him aware of all his problems that his solved at the moment of his epiphany. He recognized this elsewhere, and it might be because RJ isn't at the helm that Rand didn't include Cadsuane too.