What motivated Cersei to engage with Lancel Lannister?

To begin with, Cersei was never very loyal to Jaime, or considered him "the one". I say so because Cersei was quite taken with being the Queen and quite readily agreed to marry Robert. Now, don't picture the fat Robert showcased in the TV Series. In his prime, Robert was a famed and very handsome warrior.

From A Game of Thrones, Chapter 4

In his youth he was described as tall, broad shouldered and muscled like a maiden's fantasy. He kept himself clean shaven. However, after winning the crown he became overweight from excessive feasting and drinking, eventually gaining over eight stone in weight and growing a beard to hide his multiple chins.

Her infatuation with Robert ended quite soon when on their wedding night, Robert came on top of her whispering "Lyanna" in her ears. She was also smitten with Rhaegar Targaryen at one point of time.

From A Feast for Crows, Chapter 24

When she had been presented to him, Cersei had almost drowned in the depths of his sad purple eyes. "He has been wounded," she recalled thinking, "but I will mend his hurt when we are wed." Next to Rhaegar, even her beautiful Jaime had seemed no more than a callow boy.

What I am trying to establish here is that Cersei didn't have eyes only for Jaime, something which the TV series might have led you to believe. But she did choose Lancel because he is said to closely resembled Jaime with his green eyes and sandy hair. There are other factors like repaying him for what he did to Robert, however, at the end it boils down to just good plain old lust!

Cersei as a character is obsessed with sex as a controlling mechanism. In the show, we only see Lancel Lannister. He is a weak willed character and easily bullied and manipulated. He is also very cowardly. Someone like him would never have had the courage to participate in the plot to kill King Robert. So Cersei uses her sex appeal to subjugate him to her will. It doesn't hurt that he is good looking and (in the books at least) somewhat resembles Jaime who is languishing in a dungeon away from a very lonely Cersei.

In the books universe we see a few other cases of Cersei using sex as a means of control. Needless to say, book spoilers ahead:

Ser Osney Kettleblack implicated Margaery Tyrell in a sex scandal in return for sleeping with Cersei. She also has a sexual episode with Lady Taena of Myr, although she is not attracted to her in the least just to get a feel to how controlling a male can be on his female sexual partner.

This is, for me, the combination of a number of factors, each one beneficial to Cersei:

• Leverage

Cersei weaponizes her sexuality, but only privately (unlike say Margeary Tyrell, who is more openly sexual and seductive). By engaging in a tryst with Lancel, she is able to manipulate and control him, guiding his actions. Boys are obviously notorious for thinking with their penises at Lancel's age.

• Loyalty

By becoming Lancel's lover, she is indicating (or mimicking) a level of trust in him, which his loyalty and chivalry dictates must be reciprocated. As Lancel is naive, he is unaware this is another form of manipulation.

• Boredeom

Cesei is naturally incredibly promiscuous, and is highly sexed. Her high sex drvie could be perhaps attached to something more than her empowering herself and being comfortable with her sexual identity: it could almost be perceived as undiagnosed nymphomania, or hypersexuality. These can be linked to Bipolar disorder, and incest is more common among nymphomaniacs.

• Substitution of Taboo

Part of the sexual rush Cesei seems to experience with Jaime could derive from its incestious nature. Perhaps engaging in this taboo with Lancel is a form of familiarity, trying to recapture this compulsion.

• Reassurance

Any doubts or uncertainty Cersei may hold about her incestious dispositions are divested when she finds another cousin who is equally open to the notion of it. She can re-assure herself that it is not such an unnatural proposition.

Mainly, so she could manipulate him. She's a pro at manipulating guys through sex (considering she can be quite stupid in politics, sex is actually her best weapon), just think about the Kettlebacks. She got Lancel (a silly boy who wanted to be Jaime) to muorder Robert - that was the main goal. Then, of course, he has the plus of being similar to Jaime. Though I really disagree when people are saying that it is about pure lust and that Jaime has never really been "the one" for her, etc. In AFFC we get to read her thoughts as she thinks that sex was ever only good with Jaime (not with Robert, not with Lancel, not with any of the Kettlebacks, not with her friend Taena). And about her infatuation on Rhaegar, it is unfair to judge her feelings towards Jaime because of what she thought when both she and Jaime were only 10 years old if I recall correctly: of course she would think Jaime a callow boy in comparison to a 17-year-old beautifull guy like Rhaegar.