[RPG] How to design a climb up a cliff challenge

TL;DR: Yes, he can attack if he has enough Movement to make the climb and does not have to use two interactions to both stow and draw a weapon.

Walls can be covered under "special types of movement." (PHB, p. 182). The rules-as-written allows that attack if he has enough movement left to climb that distance, since you can generally move and attack in the same turn.

Climbing, Swimming, and Crawling
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed. At the DM’s option, climbing a slippery vertical surface or one with few handholds requires a successful Strength (Athletics) check. (Basic Rules, p. 64 / PHB. p. 182)

If he had at least 20' of movement left, it fits RAW that he'd complete the move up the 10' wall and then attack, providing the character succeeded in the Strength (Athletics) Check you required. Since you have already determined that the wall isn't that hard to climb, there seems no reason not to allow the attack. If you had ruled it difficult terrain, he'd have needed 30' of movement to make the climb and then attack.

Other Activity on Your Turn:
[…] You can interact with one object or feature of the environment during either your move or your action. (PHB, p. 190, Basic Rules, p. 70)

What you may wish to rule on is whether or not he must

1. stow the weapon (interaction)
2. climb the wall (movement)

3. and then draw the weapon (interaction)

   If he needs two interactions (only one interaction is free) that would most often preclude the attack. (In some cases, a bonus action could still allow it.)
   You state that this wall is not hard to climb. You may or may not allow him to climb this wall with a weapon in hand. You could increase the DC of the climb if he attempts it with weapon in hand. As that wasn't specified in the question, I'm can't suggest a ruling either way.

Your Turn
On your turn, you can move a distance up to your speed and take one action. You decide whether to move first or take your action first. Your speed—sometimes called your walking speed—is noted on your character sheet. (PHB, p. 189; From Basic Rules, p. 69)

Movement and Position

You can use as much or as little of your speed as you like on your turn, following the rules here. Your movement can include jumping, climbing, and swimming. These different modes of movement can be combined with walking, or they can constitute your entire move. However you’re moving, you deduct the distance of each part of your move from your speed until it is used up or until you are done moving. (PHB, p. 190; Basic Rules, p. 70)

If the character is a Rogue with Second-Story Work (Thief Archetype) then he'd not need all 20 feet of move, just the 10 feet.

Second-Story Work
When you choose this archetype at 3rd level, you gain the ability to climb faster than normal; climbing no longer costs you extra movement. (PHB p. 97; Basic Rules, p. 28):

Creatures without hands don't need to use hands to climb.

Almost all rules for skills (and pretty much everything else) are given in relation to the PCs, who are almost always humanoid. Thus, in most circumstances the creature using the climbing skill rules has 2 arms and 2 legs and will be climbing with them.

Non-humanoid characters can't follow these rules, just like you pointed out: spiders don't have hands but can climb, so there must exist an exception for them. Since they have a climb speed they are using something other than their hands to climb. It doesn't totally work with the rules, but that is why we have GMs and not computers.

From the SRD:

• Climb Speed A creature with a climb speed has a +8 racial bonus on all Climb checks. The creature must make a Climb check to climb any wall or slope with a DC higher than 0, but it can always choose to take 10, even if rushed or threatened while climbing. If a creature with a climb speed chooses an accelerated climb (see above), it moves at double its climb speed (or at its land speed, whichever is slower) and makes a single Climb check at a –5 penalty. Such a creature retains its Dexterity bonus to Armor Class (if any) while climbing, and opponents get no special bonus to their attacks against it. It cannot, however, use the run action while climbing.

Note that with a Str score of 10 with no ranks in Climb, any creature with a climb speed can take 10 and get a total of at least 18* for a Climb check.

So according to the rules, yes, spiders and eidolons and other things with climb speeds should make checks to Climb, but in reality they would be redundant rolls because such creatures almost always have enough bonuses to their Climb skill to automatically succeed**. If circumstances forced enough penalties to their Climb, then actually rolling would be a good idea.

*For reference:

DC 15: Any surface with adequate handholds and footholds (natural or artificial), such as a very rough natural rock surface or a tree, or an unknotted rope, or pulling yourself up when dangling by your hands.

DC 20: An uneven surface with some narrow handholds and footholds, such as a typical wall in a dungeon.

DC 25: A rough surface, such as a natural rock wall or a brick wall

**Giant spiders get a bonus of +16 to climb, meaning that when they take 10 they get 26.