How do I increase my chances of finding a nametag?

Junk is any of the following items:

Bowl, Fishing Rod (Highly Damaged), Leather, Leather Boots (Highly Damaged), Rotten Flesh, Stick, String, Water Bottle, Bone, Ink Sac & Tripwire Hook

Treasure is:

Bow, Enchanted Book, Fishing Rod, Nametag, Saddle, Lily Pad, Fishing Rod (Enchanted)

With an unenchanted Rod, your odds are: 85% Fish, 10% Junk, 5% Treasure. With Luck of the sea I, it goes to 86.5%, 7.5%, 6%. Luck of the Sea II: 88%, 5% 7%, and Luck of the Sea III: 89.5%, 2.5% 8%.

If you get treasure, your odds of getting a name tag are 16.7% (1/6), so with Luck of the Sea II you have odds of 1.2% of getting a Nametag each time.

The only way to increase this would be to get Luck of the Sea III, which increases the odds to a whopping 1.3%.