If you want to challenge the Legendary Player (the only other one to complete all of the Streetpass challenges issued by Gramps), finish those challenges yourself and then talk to Gramps. Afterwards, you will be able to face your opponent, and you can fight him again after that as often as you like.
You can unlock the special Hero Mode by completing the standard game one time.
Create the Perfect Shadow Link
Using 3DS StreetPass, you can set up a Shadow Link that appears in others's games to challenge them. Your Shadow Link will be worth a bounty of rupees that your opponent can win -- if they can beat you. You can use StreetPass Setup to create the maximum Dark Link worth 999 rupees. You must possess the following:
Fully upgraded Master Sword, with 4 Master Ores
The Red Tunic found in Lorule Castle
All of the Heart Pieces and Heart Containers from all of the dungeons (except for Lorule Castle, which has no Heart Container)
Upgraded Great Spin Attack, obtained by collecting all 100 Maiamais
Two upgraded weapons equipped, such as the Nice Ice Rod and the Nice Fire Rod (for such upgrades, you must collect Maiamais and return them to Mother Maiamai)
Note that if you truly want to make your Shadow Link evil, you can equip a Blue Potion or two so that he can refill his hearts. You won't be rocking the full bounty, but you will make your opponent work for their rupees!
Weapon Upgrades and Great Spin
Collect Maimais to unlock item upgrades for 9 of your purchased items, and the Great Spin, a special, ultra-powerful Spin Attack.
Nice Bow - Bow/10
Great Spin - Master Sword/100
Unlock Super Lamp and Super Net
Beat the Treacherous Tower in Lorule to unlock the corresponding items.
Beat 10 Levels - Rupees
Beat 30 Levels - Heart Piece, Rupees
Beat 50 Levels - Super Lamp, Rupees
Beat 50 Levels Again - Super Bug Net, Rupees
Different Bat for Octoball Derby
If you equip the standard or upgraded version of the Fire Rod, Ice Rod, Sand Rod, or Tornado Rods and then start the Octoball mini-game in Lorule, you'll find that Link is using that rod as his bat, instead of a standard stick. There's no apparent impact on the actual play, however.
A Link to the Past Easter Egg
Ravio's outfit is actually an Easter egg referencing the moment when Link first goes to Lorule in A Link to the Past and turns into a pink bunny.
Wall Merging Secrets
Once Link gains the ability to merge with the wall, there are several easter eggs and silly things you can merge into. First and obviously, Link can merge into the wall in his house and slide behind Majora's Mask, making it seem as if he is wearing the mask.
Several houses in Kakariko Village have portraits and paintings in their houses that Link can merge behind. Slip behind a portrait of Makar's mask from Wind Waker, or a Cucco head, or even a Skull kid head in Thieves' Town!
The Stylish Woman in Kakariko blows Link a kiss every time he enters the house, but she's also gotten lipstick on the right wall. Merge along the wall to give Link lipstick on his face!
When entering Hyrule Castle with the Master Sword, Link can merge along the paintings retelling the events of Link to the Past. You can even take the place of the place of the Link in the painting!
Link's appearance while wall merging also changes depending on the tunic he has equipped, if he is holding a lantern, and even the border that forms as he merges changes color depending on his tunic.
Zelda NES Reference
After you complete the Thief's Hideout area, you will meet the same girl at a cave which is located southwest from Link's house. After you open the treasure chest, she will tell you to "Keep this a secret to everybody," which is a twist on the popular phrase used by a generous Moblin in the first Zelda game: "THIS IS A SECRET TO EVERYBODY."
Majora's Mask in Ravio's Shop
In Ravio's Shop you can purchase any of the main quest Items you would normally find in dungeons. Note that Majora's Mask can be seen on the wall of the shop, a reference to The Legend of Zelda: Majora's Mask.