Mechanics that don't fall under the explicitly listed categories are also feasible choices.
Please give credit to the creator of the mechanic you choose to use. Links to the card sets in which mechanics originated would also be preferred.

Below is a random assortment of mechanics generated by users on Multiverse, to save you time combing through sets.

Alex

• Backstab — Whenever this creature blocks or becomes blocked by a creature, [effect.] (Clockwork Wings)
• Connect {M} ({M}: Attach to target artifact you control. Connect only as a sorcery.) (Clockwork Wings)
• Reminisce {M} ({M}, Exile this from your hand: Return target card that shares a card type with this from your graveyard to your hand.) (Sienira's Facets)

dude1818

• Temporal — At the beginning of your upkeep, put a time counter on ~. [These cards then have an effect that relies on the number of time counters.] (The Oncoming Storm)

crimsonreaper

• Cooldown N (Exile this spell card with N time counters on it. At the beginning of each upkeep, remove a time counter. When the last is removed, return it to your hand.) (As seen on Electrozephyr.)

KeresAcheron

• Linger (When this dies as a creature, you may return it to the battlefield as a noncreature aura with enchant creature.) (Enchantment Legions
• Voidwarp (This deals damage to creatures in the form of -1/-1 counters and to players in the form of cards exiled from the top of their library.) (Voidwarped Eldrazi)

kixleart

• Ancestry (As this enters the battlefield, you may exile a creature card from your graveyard. If you do, this gets +1/+1.) (Hizoria)

Link

• Gate (Play with the top card of your library revealed.) (Anydria Storage)
• Prepare {M} (: Exile this from your hand facedown. You may look at it and cast it for {M} for as long as it remains exiled.) (Anydria)
• Scrounge (You may cast this card for its scrounge cost when it's put into your graveyard from anywhere. If you do, exile it as it resolves.) (Cards With No Home)

Morlor

• Survivor (At the end of combat, if this creature attacked or blocked this turn and isn't a survivor, put a +1/+1 counter on it and it becomes a survivor.) (Gate to Ukmor)

Samuel

• Bloodhunt {M} ({M}: Put this card from your hand onto the battlefield with a +1/+1 counter on it. Activate this ability only if an opponent controls a creature spell.) (Eragon)
• Dread — Whenever an opponent discards a card, [effect.] (Fellowship of the Ring)

Sorrow

• Hierarchy N (Other creatures you control that enter the battlefield the turn you cast this enter the battlefield with N +1/+1 counters on them.) (Foundations)
• Spasm {M} (You may cast this spell for its spasm cost. If you do, its target becomes random.) (Coaxkika)

ValleyBeyondTheReal

• Horizon (As you cast this spell, look at the top card of your library. If its a basic land, you may reveal it and put it into your hand.) (Pyrulean Daylight)

This list is by no means comprehensive! If you use a mechanic not listed here, I'll add it to the list.

The following challenges are also good places to peruse other player's mechanics: Challenge # 029, Challenge # 046, Challenge # 083, Challenge # 088, and of course Challenge # 143.

Cards With No Home and other non-themed cardsets are also good places to look.

Created Cracked Wheel, Locamotive Golem, and Overprime based on crimsonreaper's Electrozephyr.

I really like this challenge, but just can't find any ideas.

It's tougher than it looks, admittedly. I went to Cards with No Home and opened random boosters until I came across a mechanic that looked good. Figured that was the best place to start because it was the most likely place to find good abandoned ideas.

Yeah, I see now that this challenge may not have been the best idea. I tried to eliminate some of the legwork by displaying a few mechanics in my top comment, but it's still more work than the more successful challenges.

I made Wraith's Armaments and Blackened Shakujō.

I used survivor. Samoset's Disciple, Mokole-Mbembe, Millenia Redwood, Truth of Mortality