In the player's handbook there is an example in the Dexterity category of Thieves' tools, so I know that Dex is the ability that is used for ability checks with thieves' tools, but I hadn't found the ability for the other tools. Is there any info about this in the books or is up to the DM to choose the ability?
\$\begingroup\$ Welcome to RPG.SE! Take the tour if you haven't already, and check out the help center for more guidance. \$\endgroup\$– V2Blast ♦Aug 12, 2019 at 17:36
It's up to the DM to decide based on the task you're attempting to accomplish.
From the Player's Handbook, page 154, the introduction of the Tools section:
Tool use is not tied to a single ability, since proficiency with a tool represents broader knowledge of its use. For example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver's tools, or a Strength check to make something out of particularly hard wood.
In addition, Xanathar's Guide to Everything suggests that tool proficiency also includes knowledge of the related profession, further widening the potential uses tool rolls. Many of these uses are probably intelligence-related, as you use them to find hidden aspects of an object or to learn facts about its history and structure.
The DM calls for the ability check ... but there are some clues.
One of the things we discovered is the same question that you raise: other than thieves tools, what stat do I use for a given ability check when using a tool?
Proficiency with a tool allows you to add your proficiency bonus to any ability check you make using that tool. Tool use is not tied to a single ability, since proficiency with a tool represents broader knowledge of its use. (Basic Rules, p. 52)
If you dig into the various ability scores, there are some hints. For example, for gaming sets like cards and dice, we found this:
Other Intelligence Checks. The DM might call for an Intelligence check when you try to accomplish tasks like the following:
• Estimate the value of a precious item
• Recall lore about a craft or trade
• Win a game of skill (Basic Rules, p. 64)
That last item convinced our DM to use Intelligence score, not Dexterity, when we applied our proficiency in playing a dice game. (Yes, the wizard won ...). He also determined that my Fighter's proficiency with land vehicles was based on Wisdom, not Strength, since he felt that "using your judgment and experience" is the skill to draw on for using a vehicle. (Made sense to me).
Wisdom reflects how attuned you are to the world around you and represents perceptiveness and intuition. A Wisdom check might reflect an effort to read body language, understand someone’s feelings, notice things about the environment, or care for an injured person. Basic Rules p. 64)
The other guidance comes from this section of the Chapter 7 rules:
Variant: Skills with Different Abilities
Normally, your proficiency in a skill applies only to a specific kind of ability check. In some situations, though, your proficiency might reasonably apply to a different kind of check. In such cases, the DM might ask for a check using an unusual combination of ability and skill, or you might ask your DM if you can apply a proficiency to a different check. For example, if you have to swim from an offshore island to the mainland, your DM might call for a Constitution check to see if you have the stamina to make it that far. In this case, your DM might allow you to apply your proficiency in Athletics and ask for a Constitution (Athletics) check. (Basic Rules, p. 62)
The same approach can be applied to tools.
Work with your DM to arrive at a check basis that "feels" right to you both.