I think they should stick with him tbh. It's silly to hire a young manager and sack them quickly - if you want instant success/stability you can keep hiring from the treadmill of older managers who can get the job done, but there will never be genuine progression going that way. Gut instinct tells me Lampard will be a good manager in time. He's brighter than your average pro, built a career off of maximising his abilities, has a ton of experience to call upon from his playing days, close family were coaches and managers, and has a promotion under his belt.
I think he was the wrong appointment last season, probably wasn't the right manager to turn around a sinking ship, but with a fresh start he'll be fine. Everton's a mess so it's not an easy job. Even at Chelsea, he was there during their transfer embargo and did well in the circumstances. Compared to say Gerrard who is getting plaudits at Villa, but Villa weren't in bad shape when he took over and had good, mostly young players whose careers were on an upward trajectory. Fortunes change quickly in football, I'd put my money on Lampard having a more successful managerial career than Gerrard.