Why do nearby AF points look faintly illuminated in single-point AF on a Nikon D3300?

First of all: I would not be worried of other focus indicators being lit slightly, it's probably normal. As long as you can distiguish each focus point indicator I see no need to take it to the service center.

I guess you refer to the effect in the following picture, observed in the viewfinder of a Nikon F100:

The photo was indirectly taken via the mirror though the bajonett opening.

We observe that the middle focus point is active and brightly highlighted and the other two indicators are faintly lit too.

The cause is probably light of the LED lighting the center indicator being scattered around, illuminating the other points.

This type of focus indicator is probably built about this way:

(At least I would built it this way :)

• View from the side towards the edges of the display.

• 3 Layers of transparent glass with brackets "scratched" onto their surface, individually lit from the side. The light source needs to have a narrow angle. The light can leave the surface at the scratch. You may have seen exit signs consisting of a thick tile of plexiglass being lit from the side with the actual sign painted on one side. It's quite the same: the side with the plaint on it has a frosted (scratched) surface, allowing the light to leave the tile.

• To implement 9 focus points, 3 rows of LEDs for each layer would be necessary.

• Light may leak from one LED to the other sides, slightly highlighting the other layer's edges which in turn causes their brackets to be slightly lit.

I can think of the following causes of increased leaking of light:

• dust, much dust, on the side of the glass layers

• too much distance of the LEDs to the glas layers due to a severe damage