In this primitive stage of the competition, positioned amidst an unrelenting festive period, teams often opt to rotate their squad and rest key personnel.
While it's important to do this, and to also blend youth into the equation, the Toffees are without a trophy since 1995 and Moyes would be wise to plot his selection carefully.
Here's a possible lineup and some thoughts behind each pick.
Goalkeeper: Tim Howard
Moyes has a habit of playing Jan Mucha is the early stages of domestic cups, but there is little point in following that trend here.
The backup is out of contract in the summer, likely to move on and has frankly looked like a liability whenever he's appeared.
He featured between the sticks during recent cup upsets against Leeds and Brentford, and—given the circumstances—Moyes should go with his main man here.
Defence: Seamus Coleman, Phil Jagielka (c), Shane Duffy, Bryan Oviedo
Right-back has been an issue for Everton in recent weeks, with Coleman, Tony Hibbert and Phil Neville all sidelined through injury, leaving Jagielka to fill in.
Coleman should be back for this game, allowing Jagielka to marshal a young defence from his more favoured central berth.
Duffy deserves a chance to make his first start of the season and gain valuable first-team experience alongside him. Having played on loan at a decent level and featured fleetingly in the Premier League, he should be more than capable of thwarting Cheltenham's strikers.
At left-back, Leighton Baines supposedly took a fitness test prior to Everton's previous fixture, so it would makes sense for him to sit this one out. Oviedo has been a frequent substitute so far this season and deserves an opportunity to impress.
Midfield: Ross Barkley, Leon Osman, Marouane Fellaini, Steven Naismith
This is a fixture Ross Barkley must appear in to further his development, although it makes sense to play him in a wider berth, giving him more time on the ball.
In congested central areas he's sometimes looked rushed and been guilty of coughing up possession. On the rare occasions he's emerged from the bench for Everton, it's often been in a wider role.
To compensate for his relative inexperience, Fellaini and Osman would then provide the Toffees with an effective central mix of grit and guile, while Naismith would switch flanks to allow Steven Pienaar to rest.
Naismith has had limited success playing on the right and may prefer cutting in off the left, playing down a narrower, more attacking channel, with Oviedo overlapping.
Attack: Nikica Jelavic, Victor Anichebe
Apostolos Vellios is certainly a close call here, but Nikica Jelavic's lack of Premier League goals is a relevant factor. A goal and a confidence-boosting performance could ignite the Croatian and is a worthwhile risk.
Ideally, he could get on the scoresheet early and allow Vellios a decent cameo in the second half.
Playing just behind Jelavic, with five goals in 12 appearances, Anichebe is in fine form and has to start. In this system he would interchange with Naismith and Barkley, switching between areas behind and wide of Jelavic.