I noticed this post came up again so I thought I'd add to it.
What you are complaining of is Medial Epicondylitis, otherwise commonly known as "golfer's elbow", as the trail elbow in golf is under a side bending (valgus load) force during the downstroke to contact. I see this a lot in SUP paddlers that visit my clinic. What is happening is the forearm flexors - the muscles on the palm side of your forearm are being overused or over stretched or both during paddling. This can also lead excessive strain on the inside ligament of the elbow (the ulnar collateral ligament) which also serves as the attachment point for the forearm flexor muscles tendon (tendons connect muscles to bones).
Seeking treatment that involves stretching of the forearm flexors; settling of the inflammation - ultrasound and ice (tendons have a poor blood supply and as such take longer to heal than muscles and ligaments - unfortunately this gets worse as we get older 
). Acupuncture works well for releasing the tight muscles and creating a healing response for chronic (long standing) presentations.
Improving your technique as Peter pointed out is very important. Peter mentioned the thumb up idea, I'll elaborate a little further on why this is important.
The photo above shows greater strain through the inside elbow. The wrist is extended back putting greater strain (over stretching) on the forearm flexors. This coupled with the inside elbow now facing down to the water at the catch, creates excessive load at the overstretched position, which puts all the load through the tendon (mid range positions are best to accept load, not end of range)
Correcting the elbow position by trying to have the point of your elbow facing the water - not the inside of your elbow, will minimise this force during the catch phase. The net result of this will lead to your thumb on your top hand being more upright. The important point here is the position of your elbow. The second point is to ensure you are not dropping your wrist, try to keep your wrist flat so it creates a straight line across your wrist and hand - effortlessly - you should be able to do this with minimal effort not rigid.
Remember to push down at the catch as if spearing a fish with your paddle as a harpoon. Avoid 'punching' the handle forward as this leads you to drop your wrist. This is also not good for your shoulders as you are subsequently over using the top arm during the stroke to create drive. Paddle length can be an issue here. Too long a paddle will result in 'paddle punching' as you can't get 'over' the handle. I have a specific method of measuring paddle length to address this - involving shaft length.
The stretch above is an effective stretch for the forearm muscles and also assists the range at your elbow out of pronation (palm down position) which tends to hold the elbow "out of neutral" and lead to incorrect loading. If you try this stretch with both hands on a wall, keeping your palms flat to the wall and finger straight down, you will notice the injured/tight arm will sit lower on the wall than the good side. Keep stretching regularly until both are the same height.
Hope this helps, there is nothing worse than being off the board due to injury.
Cheers
Greg
Horizon Physio
