Coulson has both professional and personal reasons for wanting Daisy to lead
Whilst Daisy doesn't believe she can or is ready to lead, Coulson has always seen the potential in her to do far more than she thinks.
Coulson has been preparing Daisy for greater things since Season 1
Early on, Coulson makes it clear he's setting her up to do his (then) job:
Season 1, Episode 9
Skye: "You know what makes people tick"
Coulson: "So do you. You see the good in them, you were a friend to Hannah, you can't wait to get inside May's head, and you figured out Tobias without even trying"
Skye: "Guess I'm full of surprises"
Coulson: "But this isn't one of them. I wanted you paying close attention to the Index Asset Evaluation and Intake process because some day, you'll be really good at it - even the best"
Daisy has already held a leadership position
At the end of Season 2, Coulson assigned Daisy to put together and lead a team of her own - the Secret Warriors. Over the course of Season 3, she managed to find a group of suitable Inhumans, get them to join S.H.I.E.L.D., get them trained (where necessary), and then was able to plan and successfully execute a rescue of the main team when they had been captured by Giyera & Hive. Whilst that team was then disbanded, it wasn't due to a failure of the team or her being a bad leader, rather that Hive's sway/influence over Inhumans made it too dangerous.
Coulson had wanted Daisy to lead S.H.I.E.L.D. when he last stood down
When S.H.I.E.L.D. was preparing to go legit again in Season 4, Coulson had stood down as Director, and Jeffrey Mace had taken over. However when Daisy rejoins S.H.I.E.L.D., Coulson indicates that had she not left she would have been his intended choice:
Season 4, Episode 8
Daisy: "Plus, it doesn't hurt that the new face of S.H.I.E.L.D. is an Inhuman. Your idea?"
Coulson: "I thought the new face of S.H.I.E.L.D. should be an Inhuman. Didn't mean him though."
Daisy: Gestures at herself, surprised. "Yeah, right - maybe the comic book version"
Coulson thinks of Daisy as a daughter
We see repeatedly that Coulson is willing to go to extremes to protect and/or rescue Daisy, and that he thinks of her as the daughter he never had:
Season 3, Episode 18
Coulson: "…Daisy's always longed for a family, for a connection - I thought I could give it to her with S.H.I.E.L.D."
May: "Phil, you're not her father"
Coulson: "No, but she's the closest thing I have to a daughter"
There aren't many other choices
As stated in the other answer, he doesn't have a large number of people to choose from - Fitz & Simmons aren't remotely qualified, neither is Elena, or any of the 'lower-ranked' agents like Piper or Davis.
It's highly doubtful that May would want the job - she's repeatedly shown as being happiest Coulson's right-hand, hating anything that causes her not to take action directly, and is almost as protective of Daisy as Coulson.
Mack is a plausible choice, having been left in charge of S.H.I.E.L.D. before (Season 3, Episode 9), but his recent black & white good vs. evil viewpoint isn't really conductive to leading S.H.I.E.L.D., as the Director frequently has to work in shades of grey.
That leaves Daisy.