It's probably a cosmic ray strike, a common artifact on New Horizons images. The image, however, is of very poor quality which makes it difficult to analyse the exact nature of the spot. I'll go into this more at the end of the answer.
Cosmic ray hit
A common occurance (see below); the most likely cause without evidence to the contrary.
Background star (e.g. a nova)
Novæ (and variable stars) brighten and dim over several days. Were this a nova it would have shown in images from previous and later days.
Kuiper belt object
To not show in images from adjacent days it would need large relative motion and thus a sizable trail would be expected (none is evident).
Possible? Yes. Likely? Far, far, from it. Not worth serious consideration.
Here's the New Horizons team's standard disclaimer:
This image contains one or more objects whose brightness exceeds the detector's saturation level. This sometimes produces a "tail" of bright and/or dark pixels to the right of the object. You may also notice a faint vertical white stripe passing through the saturated object; this is an artifact called "frame transfer smear" and is associated with the incomplete removal of signal produced when the image is transferred from the optically active region of the detector to the storage region of the detector. If the target is badly saturated, you may also notice a faint, X-shaped feature nearly centered on the object; these are optical diffraction spikes.
This image contains one or more streaks associated with cosmic rays passing through the detector. Nearly every LORRI image has at least one cosmic ray strike, but most are "single pixel" events (i.e., they only appear to be in single pixel and can easily be mistaken for stars). But sometimes a cosmic ray is energetic enough that it leaves a "trail" as it passes through the LORRI detector.
Example of cosmic ray hit and saturation trails:
Source: here (via "Learn more..." link), NASA/JHUAPL/SwRI
Note that despite being a PNG, JPEG compression artifacts can be seen in this image
The frame in question:
Source: Frame K-29 of this image taken from here, NASA/JHUAPL/SwRI/Henry Throop
Enlargement below is from frame K-29 of this image
The image in question is almost certainly a stacked composite of ~10 source images and appears to have been 2×2 binned (compare here). To make faint objects more visible, the image was also likely contrast-enhanced. The image also shows clear signs of lossy compression and quantization (from 12 bits per pixel to ~5), see enlargement below. While these are (mostly) acceptable choices for the image's intended use it tends to obscure detail from the original images.
Crop showing artifacts:
Looking at the spot in question (enlargement below) a couple of things are noticable:
It is compact and relatively dim. Its overall size in pixels is about half that of the stars and of MU69. In addition, the drop-off in brightness moving away from its center is much more rapid than for the other objects. This is consistent with a bright but compact object appearing in only one image in the K-29 stack.
This object also exhibits a less-blurry structure with two distinctly brighter spots or lines. This is consistent with spray from a cosmic ray hit.
Crop showing mystery spot:
All in all, I think a cosmic ray hit is the most likely explanation. Given the very poor quality of the image in question though it's difficult to give a definitive answer. To make a proper determination one would need access to the yet-to-be-released raw imagery.
I've seen a paper with a more thorough discussion of New Horizons image artifiacts but don't seem to have saved a copy. I'll add references from it when I track it down.