I really do think side-based or go-around-the-circle initiative is best.
Important: I love the cooperation of group initiative where the PCs go in order they choose, but with seven mostly inexperienced players I think that's going to be too much debate and eat too much time. Instead, have them roll initiative as normal but that ONLY determines who goes first. Once the winner is determined, do like LordEntrails suggested- it goes around the table in a circle just like a traditional boardgame. Either clockwise every time (with the monsters going based on the DM's seat), or alternating between clockwise and counter-clockwise from fight to fight. You may also want to consider Always On initiative like Shadowdark, to impose some order on exploration activities and enforce some spotlight sharing.
Important: I agree with Tony that trying to go Theater of the Mind will be too time consuming and annoying. You want minis and visual representations everyone can refer to rather than having to repeat descriptions over and over and remind players who've had a drink or two of who is standing where every single time. I think Overgeeked is right that you don't want to get too fiddly and detailed with it, though. You don't necessarily have to do strict grid movement, but at least use minis or tokens to show the formation/general shape of the rooms, and which PCs are engaged with which monsters and standing next to which other PCs for area effects and when traps happen.
Good ideas: I think Dave2008's ideas of imposing a timer and of having players roll to defend instead of the DM rolling to attack may be good ones too. These two ideas help keep the pace up when it's a given person's turn and keeps players attentive when it's not their turn.
Less important: In terms of a timer this might also be a good time to try out a Torch Timer from Shadowdark- use an alarm with the torch burning out every hour. Maybe have the alarm go off after 50 minutes, at which point you warn the players that the torch is burning low, and then hit Snooze for 10 minutes and have it actually go out after that. If you don't do a torch timer, it could still be good and add a sense of urgency if you roll for a random encounter after a fixed period of real time. Again, have an audible alarm go off. Keep time and tension in their minds.
