I have just started working with a personal trainer and she has given me some simple seated exercises to do using light hand weights. As well as working the arms and shoulders, they also work the back and core muscles and I am finding them really beneficial.
I use 2kg hand weights - easily available on sites like Amazon (alternatively, a couple of tins of beans from your cupboard, though these won't be as heavy!)
Exercises include (in a seated position):
1) Raising and lowering the arms in a wide circle, from your hips, out to the side and meeting above your head x10
2) Rising and lowering the weights from your hips, in a straight line in front of you, to shoulder height x10
3) Starting with your weights in front of your shoulders (close to your body) and raising them to meet above your head x10
I usually do these exercises one after the other (10 reps of each, with a short breather in between) for 3 repetitions (so exercise 1 for 10 reps, exercise 2 for 10 reps, exercise 3 for 10 reps, then repeat the sequence twice more).
Remember to sit as straight as you are able and engage your core muscles for maximum benefit.
Hope these are useful