Sending resistance values over bluetooth
As you have recognised, it will need an analogue to digital converter (ADC) to read the potentiometers, and a way to package up that information, and send it over Bluetooth, in a form your phone can use. That is a microcontroller.

Reaching your size of 2x2x1 cm, is extremely challenging as a starting point. You are likely to need to develop custom electronics. I recommend you use a ready built development system, get it all working, then shrink to your target size.

For a low power application, you are looking for Bluetooth 'Smart' aka Bluetooth Low Energy (BLE).

There are specific Bluetooth 'profiles' for analogue data. There is a lot of complexity to those profiles. So if you want to take advantage of those, it is probably better to use a development system which has some support for Bluetooth or BLE profiles. 

One approach is a microcontroller with a Bluetooth Low Energy 'daughter board', for example ST Micro's Bluetooth Nucleo expansion board which plugs onto their St Nucleo-64 development boards. Several of those have low-energy microcontrollers. ST run free workshops and provide teaching material which you could probably get for free.

Several microcontroller manufacturers integrate bluetooth with the microcontroller. To make products easy to prototype, several mount the parts on modules.

One development system I am familiar with is from Cypress Semiconductor. They have two types of module a PSoC 4 BLE and PRoC BLE, along with a Development kit. Again they have free workshops and teaching material. Cypress' workshops show a demo mobile phone application which communicates analogue data, using their BLE module, to a mobile phone. ST may also have a demo mobile phone app, I can't remember if they do.

You can probably get a Bluetooth Low Energy module, which has a UART (serial) interface, but I haven't checked. Most microcontrollers have a UART and Analogue to digital converters. You will also need to understand how its 'Low Energy' features work. You will have to decide how to get the data across to the phone, and write the mobile app. 

For such a low speed application, toy might also consider WiFi. Their are many WiFi modules, but the cheapest I am aware of is Espressif's ESP8266. AGain, you will need to write the phone app.