Topical for me as I'm trying to add in some weight training. Especially for us older cyclists (40+) it has health benefits to avoiding or at least slowing the age related decline in muscle mass.
My plan is to keep weight training on non rest days for the simple reason that I believe in the view that rest days should be rest days. In order to keep it manageable I'll be doing a three day split which I've worked with before i.e. push, pull, legs/shoulders, and abs/core which can be done on any of the three days. Though the legs day includes shoulder which is a mixture of push and pull.
I'm currently following a 6 days per week cycling plan which has a full rest day on Monday and the easiest non-rest day is Friday. So I could choose to aim to make Friday the legs day then the other days which are not touching legs on harder days.
There is the case often put that you shouldn't do weights and cardio on the same day. However I was googling earlier and found this.
http://www.muscleandfitness.com/workouts/workout-tips/should-you-do-weight-training-cardio-same-day - title of the paper is "Concurrent resistance and aerobic exercise stimulates both myofibrillar and mitochondrial protein synthesis in sedentary middle-aged men."
As for what exercises - main legs will be squat, main pulling exercise is deadlift and main pushing bench press. I have noticed a benefit in terms of explosive power from doing this in the past. Downside is increased weight. However low normal BMI is not lowest risk group in older people so it might be a good thing.
Next gym visit will be Friday for legs/shoulders/core day.