First a warning, medical diagnosis is a very dificult task and a serious business. Don’t expect to condense the 6+ year training of a doctor in 32KB.
My project would be a device that can diagnose whether a person is sick by taking several sensor readings. Using that data, the device then puts the data through some algorithms to determine the sickness.
This is typical a very interesting project with far too little information to give a meaningful answer (my opinion) So to get the requirements a bit right a few questions:
What is the goal of your project? a sellable device or a proof of concept?
Can you tell more about the sensors you have in mind?
Can you tell more about the algorithms you have in mind?
What are your ideas of the measurements? invasive, skin or remote?
What is your idea about the frequency of the measurements?
What is your idea about the duration of the process/analysis?
Can you define what sicknesses your device should be able to recognize? Or just yes / no?
I am completley new to the Arduino, but I think it will work for my project.
How can you draw that conclusion when you are completely new? Or it implies you have a reasonable elaborated design in mind …?
Can you tell us more about your software and hardware skills ? What level? (biometric / medical skills)
What have you tried and investigated so far?
Do you have a team? What is their knowledge level? (also biometric / medical)
So far first round of questions, now some answers (to keep yin yang in balance
with some URL’s to get started.
I would measure the following:
furthermore to interpret some of theses measurements you need to know if the patient is (fe)male, his or her age, known medical conditions e.g. allergies! eating habits, time of the day, current stress levels (e.g. due to work), … For a lot of things there is no direct sensor only a list of Q&A.
After doing this quick scan I think you definitely can make a bio-measuring device with an Arduino but that it is not trivial to make a diagnose.
I certainly would not give another diagnosis than “all parameters are within normal limits” and “go see a doctor”.
For esthetics I recommend a graphical screen to show all values at once. Finally most of the parameters mentioned above need to be monitored over a certain time (maybe even a day) so logging to an SD card should be considered.
Hopes this helps you to get started! (anyone can 
Succes!
< all disclaimers apply>