I had a player who wanted to help someone else stand up before his turn starts so he can reach the enemy. Is this written in the D&D 5e rules? If not, how would you rule this? I am thinking of using the helping character's movement instead.
Yes you can
PHB Chapter 9: Combat, page 193, emphasis mine:
Your character can do things not covered by the actions in this chapter, such as breaking down doors, intimidating enemies, sensing weaknesses in magical defenses, or calling for a parley with a foe. The only limits to the actions you can attempt are your imagination and your character's ability scores. See the descriptions of the ability scores in chapter 7 for inspiration as you improvise.
When you describe an action not detailed elsewhere in the rules, the DM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure.
D&D is not like a computer game. Your player can announce various things, including "I'm trying to help him stand up". It is your job as a DM to make a ruling instead of just saying "no you can't do this because, you know, this action isn't in the list".
RAW, the PC spends their Action trying to do this. You might ask the player to make an ability check, if there is a chance of failure:
Actions in Combat
When you take your action on your turn, you can take one of the actions presented here, an action you gained from your class or a special feature, or an action that you improvise.
When you describe an action not detailed elsewhere in the roles, the DM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure.
I'd say it should be an automatic success in this case — the action itself is trivial, and spending an Action in combat is already a decent price. Just keep your players informed: "Yes you can help him, but you won't have time to do anything else in this turn. Will you do this?"
It should not go for the price of just the helper's movement thought — firstly, because it contradicts the rules (you must spend an Action and you can't really "transfer" your movement feet to another character), and secondly, it creates a precedent of doing something useful in combat almost "for free" (you have plenty unused movement in combat usually).
Working together by taking the Help action (PHB p.175 and 192) provides Advantage on one ability check or attack roll. Since standing up from prone does not require a roll, this makes no difference. If the DM rules that because of the circumstances standing up requires an ability check to perform (e.g. on ice), you could help with that.
Making a ruling
Of course, the DM is free to make a ruling and allow it. If the prone character would take the Dash action, they could cover the same distance as when they start standing upright. So either way it took one action from the party to achieve and thus would not disturb the action economy much. In certain circumstances I would recommend asking for a check to perform this, most likely a Strength check from the helping character. Imagine a weak halfling trying to drag a goliath 5 times their weight up from the ground. Also if standig up would require a check already, Help should work as normal and just provide Advantage, instead of succeeding automatically.
Unopposed grapple, then move the target to standing (which move requires half their move speed). One could argue that it requires the "helper" use half the target's move speed doubled (per moving a grapplee); I've always ruled that the grappler isn't really moving with the grapplee during this maneuver and so the grappler must "spend" half the target's move speed to right them, but not any movement on their own behalf.
In other words, I allow PCs to move to a prone target and right them as long as the helper has half the target's move speed and the grapple-instantiating attack left to spend. But an argument could be made for it requiring a grapple-instantiating attack and the full move speed of the target. This GM finds that too costly for extending down a hand to an ally, but I've heard others argue the other way.