My wizard has access to a new magic item that allows for a once a day concentration on two 3rd level or lower spells and am looking for some suggestions on how to use this in interesting ways. Also, we're still running 2014 rules.

Ring of the Split Mind

Ring, very rare (requires attunement by a spellcaster)

This crimson ring is split into two half-circles that magically hover a hair's breadth apart. While wearing the ring and concentrating on only one spell, you have advantage on Constitution saving throws that you make to maintain your concentration on that spell when you take damage.

 

If you're concentrating on a spell of 3rd level or lower and wearing the ring, you can choose to cast another spell of 3rd level or lower that also requires concentration without losing concentration on the first one. While concentrating on two spells in this way, make a Constitution saving throw for each spell as normal whenever you take damage to maintain your concentration. When you do, roll the d20 for each spell's saving throw in the order in which you cast them. You are vulnerable to psychic damage while concentrating on two spells in this way. Once this property of the ring has been used, it can't be used again until the next dawn.

Hold Person + Cloud of Daggers

Summon Shadowspawn + Summon Undead

Slow + Web

Haste + Fly

Directly from the DMG

Beware of adding anything to your game that allows a character to concentrate on more than one effect at a time [...]. Rules and game elements that override the rules for concentration [...] can seriously unbalance or overcomplicate your game.

Whether it actually breaks things comes down to how you use it but it’s such a bad idea the DMG explicitly calls it out.

Idea: haste + haste again.

Another idea: slowness + slowness again 

Better yet, Haste on the Barbarian, Slow on whatever the Barb is beating up.

While a nice perk, this isn't going to be able to achieve anything that two casters working together wouldn't otherwise be able to achieve.

1) Many concentration spells use a bonus action on a following turn, so they would conflict with one another.
2) Any spells with a 1 minute duration get a bit shafted due to wasting rounds of combat to bring them online, so you'd want to use, at max, one of these.
3) Any, non-self, Single Target spells could be replicated with Twin spell metamagic.

For a half-caster, something like stacking Hex and Hunter's Mark for 8 hours would be great.

For a Druid, this would be fantastic to maintain effects while in Wildshape.

My first instinct would be to Dualcast Summon Fey, and have they spam Fey Step (Mirthful) to essentially give every NPC disadvantage against being charmed.

Second instinct would be to cast something like Ashardalon's Stride + Invisibility. Since passive damage spells don't need to be recast or activated with an explicit attack, you can arguably damage opponents without losing invisibility.

Quote from Memnosyne >>

Second instinct would be to cast something like Ashardalon's Stride + Invisibility. Since passive damage spells don't need to be recast or activated with an explicit attack, you can arguably damage opponents without losing invisibility.

The 2024 version of invisibility closes that loophole, unfortunately

The spell ends early immediately after the target makes an attack roll, deals damage, or casts a spell.

Fortunately, OP specified that they were sticking with 2014 rules.

Quote from Memnosyne >>

Fortunately, OP specified that they were sticking with 2014 rules.

Yeah, I was lamenting the loss of a fun combo, not trying to correct you