There are a few possibilities but its fairly easy to isolate the likely problem.
Unclip
the noodle (where the cable comes out) and check if the problem is in the V brakes or the cable.
If its the V brakes, it could be a broken spring, or the brakes need oiling.
If the brakes are free to move and spring back firmly, its likely a problem with the cable or the brake lever.
Without pulling the cable out of the housings, slide the cable back and forward - it should be free to move.
If not, check the brake lever is free, or it could be a corroded or kinked cable.
If its the cable, a replace is only a few dollars and is the best option (I almost always put in new outers as well).
You can try adding some lube to the cable and working it, its usually successful but never as good as new cables.
If the cable is kinked, you may have luck straightening it, but again, a new cable is always a better option.
bicycles.stackexchange.com/users/3924/mattnz
Daniel's comment is right, in addition check the brake lever for stiffness.
You should be able to undo the noodle from the noodle-holder, the brake arms should extend out to the sides, and each should have about the same hand-pressure required to close up.
There should be a "tension" adjust screw in the base of each arm, although I have bikes with just one and another bike with none.
Each brake arm has a spring which can normally slot into one of three positions too.
It could be one of your springs has got really weak
, they don't tend to get stronger with age.
Its a lot of trial and error and tweaking.
bicycles.stackexchange.com/users/19705/Criggie
Another crude (but often effective) hack / adjustment for older or ineffectual springs is as follows: the springs - which are essentially just a long piece of heavy gauge wire that gets tucked behind a protruding peg to hold them in place - can be "untucked" (for lack of a better technical term) and then you can manually bend them out slightly by using a screwdriver, e.g. behind the curvature of the spring arm, for leverage.
This is an admittedly rather crude but
yet still often very effective "hack" to bring them back to life, provided you don't overdo it to the point of causing more harm than good.
This would be a hack to employ after other more "proper" ways of adjustment (e.g. tension screws or what not) have been attempted first.
bicycles.stackexchange.com/users/29861/Steve Harrington
I had a similar problem in the front brake of a Schwinn Loop folding bike, but the arm wasn't moving at all.
It seems like it's all rusted inside, because I couldn't even take it off even after soaking it in WD-40.
I did manage to move it a bit side to side, but with a lot of force.
I'll try heat from a hairdryer in a couple of days when the new brakes arrive.
But if it's stuck like that, you might need to change the brakes.
bicycles.stackexchange.com/users/44060/Sebastian
I had one of my v-brake arm sticking to the wheel this morning; I wanted to get to work on my bike
so this is what I did: Remove the arm, plunge the base for 5-10 min in ultrasonic bath with degreaser solution (if you don't have one, just immerse it while moving the spring hub).
On the spring side, drip WD-40 on the hub while moving it until it comes out clear on the other end.
Note that on top of cleaning it WD-40 also removes the water contained in the degreaser solution so it does not rust afterward (hint: WD-40 stands for Water Displacement, 40th formula...)
At this point the hub should move with very little resistance.
Wipe off and put thick oil on the hub while moving it so it penetrates.
Reinstall and adjust screws to get even tension on both arms.
NB: I did not even bother removing the pad, I only immersed the bottom end with the moving base.
Just try not to put any oil on the pad as it'll weaken it.
bicycles.stackexchange.com/users/14482/Thomas Guyot-Sionnest
I have a folding bike with v-brakes and I noticed that when I pull on the rear brake lever, one arm of the v-brake doesn't go back to its original position. I tried adjusting the screws but it still doesn't go back. Although I'm not entirely sure if, when it stays in place, it hits the rim.  What could be the likely cause of this? The spring? Lubrication? Thank you.
V-brake arm not returning to "open" position
bicycles.stackexchange.com
<brakes><rim-brake>
bicycles.stackexchange.com/questions/43157
bicycles.stackexchange.com/users/29379/dork
If the brakes are free to move and spring back firmly, its likely a problem with the cable or the brake lever. Without pulling the cable out of the housings, slide the cable back and forward - it should be free to move. If the problem is the cable, it is best to replace it and does not cost much. It could be a weak spring. Another option is to try lubricating or WD-40.  The spring could be broken or stiff. Both sides should extend outwards about the same distance when the noodle is removed, and should provide similar resistance to closing. The tension screws can be adjusted on either side and the springs should be cleaned of dirt and grime. If the springs are working correctly, check that the cable can move freely in the housing and the brake levers are not stuck. If the cable is worn, kinked or otherwise stiff, replacing the cable and outer sheath may be the best solution.
The spring could be broken or stiff. Both sides should extend outwards about the same distance when the noodle is removed, and should provide similar resistance to closing. The tension screws can be adjusted on either side and the springs should be cleaned of dirt and grime.  If the springs are working correctly, check that the cable can move freely in the housing and the brake levers are not stuck. If the cable is worn, kinked or otherwise stiff, replacing the cable and outer sheath may be the best solution.