Is there a Free-To-Play item that my character can hold in Runescape that increases experience output, even slightly?

Solution 1:

You can have urns in your inventory that will grant bonus experience in fishing, cooking, mining, smithing, woodcutting and prayer. The lower level urns are free to play items.

Also, cooking on a bonfire will slightly increase the cooking xp gained, as will burning logs on a bonfire instead of burning them normally. The lower level bonfires are free to play.

Solution 2:

For Runecrafting, you can use one of the three Runecrafting Gloves to double the amount of XP earned when crafting runes of the glove's type.

They have a limited amount of uses, and can only be obtained from the Fist of Guthix minigame, but are, in my opinion, worth the cost.