Strength/ weight training absolutely plays an important role. How much of a role it plays, however, as well as exactly what it entails, will depend on the age and level of physical development of the player.
A properly designed strength training program that targets the muscles of the posterior chain (glutes, hamstrings and spinal erectors working together as a unit), as well as various unilateral (one sided ) exercises to help improve balance, stability and coordination, can do wonders to help guard against ACL injuries.
That said, proper attention must also be paid to dynamic warm-ups, instilling good running/ change of direction, mechanics and encouraging your athletes to stretch regularly and use a foam roller. As you can see, that can become a bit time consuming.
In terms of trying to establish a balance between actual lacrosse training and the types of things listed above, it's difficult to quantify and really depends on the age and ability level of the players in question.
Generally speaking, older more advanced players will be devoting more time to lacrosse with travel teams, school teams, camps, clinics and such. That combination can result in a lot of overuse of specific muscles and movement patterns- which makes strength training (as well as the other variables listed above) that much more important.
It's also a good idea to get younger players used to taking care of their bodies at an early age as well though. Granted, they may not be lifting weights and such, but doing body weight strengthening drills and working on developing good running mechanics, balance and coordination will only help them as they get older.
While each player is different, I would say that at the high school level 2-3 total body strength workouts per week would be a good rule of thumb. These could either be combined, or alternated with a couple of sessions specifically dedicated to speed and agility training. And of course, daily stretching and foam rolling.
Younger players can get by with a couple of workouts per week that focus mainly on body weight strengthening and then add in light resistance in the way of medicine balls, dumbbells, resistance bands etc., as they get stronger.
Thanks for the question. Let me know if you have any follow ups. I absolutely love this stuff and would be happy to help out!