The DM demands the ability check … but there are some hints.
One of the things we've discovered is the same question you posed: what stats do I use, other than theft tools, for a particular skill check when I use a tool?
If you are familiar with a tool, you can add your competence bonus
every skill check you perform with this tool. The use of tools is not tied to one
single ability, since the competence with a tool is broader
Knowledge of its use. (Basic Rules, p. 52)
As you study the various ability values, there are some pointers. For example, for game sets like cards and dice we found:
Other news reviews. The DM may request a secret service check
When you try to perform tasks such as the following:
• Estimate the value of a precious item
• Remember traditions about a craft or a trade
• Win a game of skill (Basic Rules, p. 64)
This last point convinced our DM to use intelligence points, not dexterity, when we used our skills in playing a dice game. (Yes, the wizard won …). He also found that my fighter's ability to handle land vehicles was based on wisdom rather than strength, because he believed that "using your judgment and experience" is the ability to use when using a vehicle can. (Made sense to me).
Wisdom reflects how much you are attuned to the world around you and
represents perception and intuition. A wisdom check might reflect
an effort to read the body language, to understand the feelings of someone, to notice
Things about the environment or the care of an injured person. Basic rules p. 64)
The other guide comes from this section of Chapter 7 rules:
Variant: Abilities with different abilities
Usually, your ability training is only for a specific type of ability review. In some situations, however, it may make sense for your knowledge to apply to a
different kind of check. In such cases, the DM may request a check with an unusual combination of abilities and skills, or ask your DM if you can apply a skill to another check. For example, if you need to swim from an offshore island to the mainland, your DM may request a constitution review to see if you have the stamina to make it this far. In this case, your DM can enable you to apply your athletics skills and apply for a constitutional (athletics) review. (Basic Rules, p. 62)
The same approach can be applied to tools.
Work with your DM to get a check basis that "feels right" to both of you.