As with purchased clothing, stretch can be your friend when sewing for yourself. All kinds of jersey and knitted fabrics plus wovens with some stretch can be good, especially when small alterations are made to the pattern before cutting out.
Again as with RTW some pattern styles are more sympathetic to larger arms from the start. Kimono styles, flared or bell sleeves etc. However do be careful to measure the pattern piece in the same place as where you arm is wider to make sure it fits well. I remember making a tunic with bell sleeves which was loose on the lower arm but far too small on the upper arm. Hopefully I'd know better now having learned the hard way.
I am going to dig out some of my 1990s patterns which had more wearing ease and might well have looser more comfortable sleeves.
I'm going to start with the cheats and fudges before getting into the proper alterations. I have got away with many of these over the years when I did not know how to do the other alterations. The plus with lots of these is that they are easy and don't involve hacking the paper tissue about too much.
Use a bigger sleeve. If you are making the size 16 in a multisize pattern, add the size 18 sleeve. On a knit garment you can usually ease the slightly larger sleeve in without it being a problem. On a woven garment, cut the armhole to the larger size to insert the sleeve.
Use smaller seam allowances at the part where your arm is fatter, tapering to the normal seam allowance in the lower arm, and right near the armhole.
Use a pattern where the sleeve has gathers, darts or pleats in the sleeve head. Use a larger sleeve and have larger gathers at the top of the sleeve (only good if you like that look).
When cutting out (or you can alter the tissue slightly) starting at the armhole cut a very slight curve away from the pattern tissue tapering back in again before the wrist.
(I often combine 1, 2 and 4 to get the equivalent of 2 or 3 sizes larger at the bicep) .
Use a 2 piece sleeve and taper out slightly on both seams. This adds in more places so is great if you need a lot of extra adding.
IMHO a shaped 2 piece sleeve with extra added looks the best and hangs nicely as you can more appropriately reflect the arm shape. It is sort of like princess seams for arms.
You can also create a 3 piece sleeve which gives even more seams those these are harder to find.
The most popular method is something called 'slash and spread'. This really needs a picture so rather than steal one from another site I will add a few links in.
It basically involves drawing two lines - one horizontally across the sleeve and one vertically down the sleeve - cutting them open and gently pulling the sides apart to make extra room within the sleeve. You then need to smooth off the curve at the top and fill in the hole with tissue.
I tried just cutting the lines and spreading out each time I used the sleeve pattern, but eventually it ripped at the seam allowances and fell apart, so don't be lazy like me, add that tissue!
I have two frustrations with this method.
1. If you fully redraw the sleeve cap afterwards you make the sleeve cap longer, needing a larger armhole, so you are in essence just putting the larger sleeve in anyway.
2. None of these variations change the outer curve of the sleeve, which matters to me as the extra is on the underside of the arm in my case (your arm shape may vary).
For all of the above I have assumed some sort of set in sleeve but you can do something similar with a raglan sleeve and just cut the sleeve wider if it is a cut on sleeve.
These alterations will just make a sleeve big enough for to get a larger arm into the garment. There are lots more fitting adjustments for forward shoulders, rounded or square shoulders etc as well as options for shaping the sleeve itself to hang better from the shoulder. However I still think these simple things are worth trying and hope they may help.