Dragonborn, as I understand it, are considered to be one of the weakest, if not the weakest, playable race. Their flavour is cool, especially for people who like dragons, but their being sub-par in terms of balance can be off-putting.

Furthermore, dragons all have darkvision (by which I mean "true dragons", those listed under the "Dragon" section in the Monster Manual), whereas dragonborn do not, and this has always bugged me. I know that dragonborn are humanoids with dragon heritage, but if their heritage is strong enough to give them scales and breath weapons, surely they can see in the dark?

Anyway, the point of this is that I am considering adding a darkvision racial trait to the Dragonborn playable race (specifically, the standard 60 ft. of darkvision as per most of the other playable races). How would this affect balance? I doubt it would overpower them, since darkvision isn't that strong, but where does this see them when compared to other races?

I don't necessarily want a ranked list of all playable races, but would this change see the Dragonborn race still quite underpowered, about average and on par with some of the better balanced races, or does this push them into the above-average-but-still-not-overpowered, such as the Mountain Dwarf? In other words, I want to know if this makes the Dragonborn race more attractive to players without making them too much so.

Very Little

The races in 5e are very closely balanced and you need to take into account what the race already has instead of darkvision when comparing it to other races with darkvision.

That being said, I personally don't think darkvision is all that useful and depending on your setting and what adventures you go on, it could easily never come into play. Also, depending on your party makeup the DM might make decisions on how darkvision works to keep the playing field more even.

Whilst it is purely a fan-based piece of work, I have found this guide to homebrewing races to be useful in the past. They have some good ideas (as well as a very good creation guide) and state both that Dragonborn are one of the weaker races and that darkvision is one of the lighter additions, value wise. On that basis, adding Darkvision to a Dragonborn would still put their value to below average.

It'd make them more balanced with other races

Detect Balance is a fairly popular guide for homebrewing races and determining whether they are balanced with one another. It includes scores for the official races and breakdowns of how those scores are determined. It also assigns values for various possible racial features.

The second tab of the sheet provides some info about the sheet and how it's meant to be used:

The average score for PHB and EE races by this scale is about 25.

The recommended score for new homebrew races is 24 to 27. Races should generally not eclipse 30 or fall below 20.

Values are approximate. If a race is few points higher than another that's not statistically significant. It doesn't mean it is better. Depending on circumstances it may not perform as well. Every campaign is different, what turns out to be useful will vary a lot.

The scoring unit is in quarters of ASIs, or Ability Score Increases. In other words a feature worth 4 is approximately equal in value as being able to raise an attribute by 1.

The third tab of the sheet has a graph comparing the official races at the top, followed by the individual breakdowns for each race. Dragonborn are listed first, being alphabetically the first of the PHB races. They're given an overall rating of 21 points. Their racial ability score increases of +2 (to Strength) and +1 (to Charisma), totaling +3, are obviously given a rating of 12 points. Their breath weapon is rated as 6 points and their damage resistance is noted as being worth 3 points, with the note:

The breath weapon and resistance are nicely balanced-- If you pick rarely resisted damage for your breath weapon, you likely have a resistance to an uncommon damage type.

These ratings generally match those described on the first tab of the sheet.

Now, let's look at the point ratings assigned to darkvision on the first sheet. 30 feet of darkvision is given a rating of 2 points, 60 feet of darkvision is worth 3 points, and 120 feet of darkvision is 4 points. If we add the typical 60 feet of regular darkvision to dragonborn, this brings their total point rating up to 24 points - still well in the middle of the usual range for most races.

By comparison, the various dwarf subraces (plus base race) have 29-30 points; the elves have 27-28; gnomes and halflings have 23-24; half-elves have 29; half-orcs have 24; and tieflings have 23 points. (Humans are outliers; regular humans have 17 points, while variant humans have 33 points.)

Dragonborn are normally on the lower end of the typical range, and adding darkvision brings them back to about average.

(Of course, any such rating doesn't mean much in a vacuum; certain low-rated races might be better for certain builds, and certain highly-rated races might be worse for other builds.)