Inner elbow/bicep tendon pain from drop bars?
Tendonitis (tendonosis) is one possibility, however I would expect some pain during riding especially during the beginning of your ride. When you say biceps tendon I assume you mean near your elbow.

That said, the first things I would investigate are ulnar nerve irritation or compression at the cubital tunnel a/o the ligament of Struther's. The ulnar nerve weaves through several fibrotic "soft" tissues in the upper arm where entrapment can occur including the medial portion the triceps. The vibration of the road (absorbed by hand, triceps, and FCU muscle mentioned below) coupled with wrist extension and wrist radial deviation can put sustained tension on the ulnar nerve which can cause entrapment and damage.

The flexor carpi unlaris muscle can also contribute or cause this problem.  Repetitive inflammation of any tissue in the body results in the development of scar tissue which can occur between muscles, fascia, and within the sheath through which large nerves travel. As scar tissue continues to develop, the restrictions or fibrosis will increase in these areas.

Go see a LMT or chiropractor who is certified in Active Release Techniques so they can help break up the scar tissue and get that nerve moving more freely.  Surgery should be a last resort in this and most other repetitive use musculoskeletal injuries. This is related to and can be exacerbated by cyclist's palsy, and Tunnel of Guyon entrapment of the ulnar nerve. I experience this problem as well, mine is worsened by core weakness and too much pressure on the hands. Higher handlebars will help get some pressure off your FCU and triceps as well. They are ugly as sin, but those long tall quill stems are helpful. 

Ride On.

Eric Poehlein DC