Here;s a list from Lan Sluder,

Tina's, Caye Caulker --

Blue Wave Guesthouse, Caye Caulker

Falconview, Santa Elena

Cayo Backpackers Hostel, San Ignacio

Pedro's Backpacker Inn, San Pedro

Nature's Way Guesthouse, Punta Gorda

Depending on how you define hostel, you could also include a number of other places, including

The Trek Stop in Succotz

Amigos on Turneffe atoll

Glovers Reef Resort on Glovers atoll

Two or three of the places on Tobacco Caye, including Gaviota and Tobacco Paradise

Hummingbird HideAway, near Corozal Town

Spiritwind Reef Resort, Dangriga (the dorm section)

BFREE, Bladen, Toledo

Paradise Vacation Hotel, Placencia village