For long-term outdoor use, a normal paper print in a household-style frame is not a good choice. Better options mentioned by the community are:

• porcelain/ceramic memorial photos: the image is fired in a kiln, so the colors are much more stable in sun and weather than ordinary photographic prints.
• photo-etching on metal: a durable option often used for memorials.
• laser engraving or etching directly into polished dark stone: this can create a permanent image, though very fine detail may soften over time with weathering.

If your priority is resisting fading, a fired porcelain/ceramic memorial photo is likely the strongest match because kiln-fired inorganic pigments hold up well outdoors. Metal photo-etching is also a practical durable choice. If you want something integrated with the monument itself, ask a monument maker about laser-etched or engraved stone portraits.

Whichever route you choose, memorial items outdoors will still age, so having a duplicate made for future replacement is wise.