For all those who are putting together potential systems together for the OP, the system requirements for sims3 taken from steam are:
# OS: Windows XP (Service Pack 2) or Windows Vista (Service Pack 1)
# Processor: (XP) 2.0 GHz P4 processor or equivalent; (Vista) 2.4 GHz P4 processor or equivalent
# Memory: (XP) 1 GB; (Vista) 1.5 GB
# Graphics: 128 MB Video Card with support for Pixel Shader 2.0
# DirectX®:
# Hard Drive: At least 6.5 GB of hard drive space with at least 1 GB additional space for custom content
# Sound:
# Built-in Graphics:
* Intel Integrated Chipset, GMA X3000 or above.
* 2.6 GHz Pentium D CPU, or 1.8 GHz Core 2 Duo, or equivalent
* 0.5 GB additional RAM
If thats all the OP's sister plays thats not that demanding even with expansion packs added. My daughter played sims years ago and it would run on anything reasonable.
The cheapest setup will run this game well. A cheap-as-chips AMD rig would run this game for years to come. 8gb ram, nope: fancy cooler, nope: ssd, don't make me laugh.
OP if you don't mean the game called the sims then my apologies, also if the pc is used for more demanding work that you have not mentioned.
Your right, a cheap of the shelf system would be fine if it the game called sims. Overclockers have a few systems that are cheap, add in a GPU and your good to go.
Edit:
King Damager's spec is also very good, change the ram to 4GB, maybe get rid of the gaming keyboard for a cheaper one and the cooler if one comes with the chip, and your withing budget, add a GPU at a later date if usage becomes more demanding.