As far as i am aware we should not be looking for the meaning of mantras. We should chant the mantras that we fell we are attracted to. Mantras are chanted to kindle the spiritual enerfies with us. They make a positive effect on our mind and heart and brings us nearer to soul.
So when we feel a soothing and deep effect, or if we feel the spirit within us is evoked or when the hairs of our soul stands on their on their end, we should repeat that mantra more and more.
It is more important to quench our thirst by drinking water than to know more about water. Reading a lot abiut water, knowing its chemical formula like H2O will not satisfy the thirst. it is ok if after satisfying out thirst by drinking water we drink may try to find the meaning of what water is, still drinking of the water is always of more importance than finding it's meaning or gathering much information of the water.
Sai Baba used to ask devotess Naam Saptah. Any divine name of God or mantra would be chanted continuously for a week. By this Sai Baba also indicated that the effect of chants gather more spritual energy when it is being done on continuous basis. The dynamo of the mind should churn more and more chants to create that energy just as a dynamo has to be spin more to create more electric energy . Hence Shraddha and Saburi. We cannot except to light our house after an hour of dynamo spinning.
Now coming to the meaning of Om Sai Ram. Om has been given many meaning yet it is beyound all meanings. The rishis and sages who went deep within their self experiences this sound AUM
ॐ and it has great spiritual significance. Again it has to be experinces. We can expereince it. Just sit with your body relaxed and straight, mind also relaxed after taking a slow deep breath chamt AUM, and you will feel the effect it creates on mindand body. Every day if the chant is repeated it reveals ots new meaning.
Sai is the name of God given to Sai Baba of Shirdi. In Sacred Text Shri Sai SatChritra Chapter 3. In Sai Baba's saying one can note that Sai Baba said that even the chanting of name Sai with Shradha (Faith) will burn away the sins (the sanskaric effect one reaps of hurting other beings) of speech and deeds.
Ram. Ram again is a spiritual name with great powers. Many devotees has felt and still feels the power of name Ram (Raam). So this manta Om Sai Ram combines the power of three powerful chants.
About the chanting of the mantras one thing comes to my mind what Sri Ramakrishna Paramhamsha said about it. In the bookGospel of Sri Ramakrishna it is recorded while His talks with one devotee that The chanting of Mantra has great effect and leads one to the God Sat Chit Anand. It bears fruit even if the effects does not show immediately. It is like a seed that will germinate, the tree will grow and bear flowers and fruits.
However the effect is more powerful when one has his mind on the mantra and God. The unconscious repetition of the mantra while the mind is engaged in the worldly affairs has little effect. ~ Om Sai Ram