Q: Why do my mother's contacts appear on my iPhone?

Are you sharing an Apple ID? If so, one of you should get a new one.

As Meg stated, but see the following:

Using More than One iDevice on the Same Computer

This applies mainly to couples who are adding another device and do not want their email, messages, etc. being duplicated on both devices. To begin read: How to use multiple iPhone, iPad, or iPod devices with one computer. You need to establish a separate Apple ID and password for whomever will use the new iDevice. See Apple - My Apple ID and Frequently asked questions about Apple ID. The easiest way is to do this on the computer using iTunes: iTunes- How to set up an Apple ID within iTunes.

On the computer create a new user account for the person with the new iDevice. This will be the user account that person will always use. He/She will no longer use the other user account. This way that person will have a separate iTunes Library. Note that once done you cannot "share" apps under a single Apple ID. Each of you must purchase your own apps under your own Apple ID. When you decide to split up like this one of you will need to re-purchase the apps that they want to use.

Start by transferring the new device(s) to a new account along with all your data.  Save any photo stream photos that you want to keep to your camera roll (unless they are already in the camera roll) by opening your Photos app, tap on Albums icon at the bottom. Now, tap on My Photo Stream album; tap Select; tap on the photos you want to select;, tap the share icon (box with upward facing arrow) in the lower left corner; then tap Save to Camera Roll.

If you are syncing notes with iCloud that you want to keep then you need to open each of your notes and email them to yourself. Later you can copy and paste the text into new notes created in your new account.

Tap on Settings > iCloud > Delete Account (only deletes it from this device, not from iCloud; the person keeping the current account will not be affected,) provide the password to turn off Find My Phone and choose Keep on My iDevice when prompted.  Sign in with a different Apple ID to create your new account. Choose Merge to upload your data.

Once you are on separate accounts, you can each go to icloud.com and delete the other person's data from your account.

Note: The essence of the above was created by user, randers4. I

have made substantial changes to improve readability and syntax.