What is the history behind the name
Why is it said that to keep idol of Lakshmi Ganapathi inside the house such that Ganapathi won't allow Lakshmi to leave the house? What was the history behind that?
I happen to search Internet and the first few Google results seems to lead to this text in this site.
Why Lakshmi Ganesha worshiped together? On the day of Diwali Lord Ganesha is worshiped with Goddess Lakshmi instead Lord Vishnu. There are many questions arises, why Lord Vishnu is not worshiped with Goddess Lakshmi and what is the relation between Lakshmi and Ganesha? The answers of all these questions can be answered by one mythological story. Mythological Story Once upon a time Lord Vishnu and Goddess Lakhmi were having conversation in heaven and Goddess Lakshmi was praising herself and telling Lord Vishnu that she is the most worship worthy in the world and by her grace, a person can get all the pleasures of this world and becomes happiest person. By hearing the self praising by Goddess Laksmi, Lord Vishnu said to her for reducing her ego, “You are having all the qualities, still you did not feel the joy of motherhood till now and for a lady, the motherhood joy is the most important thing in this universe.” The comment from Lord Vishnu made Goddess Lakhmi become disheartened and in this pain she went to her friend Goddess Parvati. After listing the problem of Goddess Lakhmi, Parvati asked her, “How can I help you?” Goddess Lakshmi said, “You are having two sons and if you could give me your one son you will still have one son and I could get the grace of motherhood. So, in this situation you can help me out.” By listing her Goddess Parvati said to her, “I'm having two sons Kartikeya and Ganesha. Kartikeya is having six mouths (Faces) and because of this he needs to eat all the time. My second son Ganesha is very naughty, if I missed to keep eye on him for a bit, he ruined everything. And you cannot stay at one place for a long, so tell me how can you take care of my sons? By hearing this Goddess Lakshmi said to her, “I'll keep your sons close to my heart and will shower my all love on them either Kartikeya or Ganesha. I can take care both of them. All the servants of heaven will serve them day and night so please give me one of them as my adopted child.” Maa Parvati knew her both the sons very well so she gave Lord Ganesha to Goddess Lakhmi as her adopted son. Goddess Lakshmi became very happy and said to Goddess Parvati, “From today onwards I'm giving my all accomplishments, luxury and prosperity to my son Ganesha. Also, Riddhi and Siddhi Lord Brahma’s daughters are alike my daughters will be married soon for which I give my word to you. I will fulfill all the desires of Ganesha. In all the three loks which person shall not worship Lord Ganesha and do back biting, I will be miles away from him. Whenever my worship will be performed, Lord Ganesha’a worship will be a must. Who will not worship Shri Ganesha along with me, he cannot get Shree or myself. By listening this Maa Parvati become too much happy and handed over her son Ganesh to Laxmi Ji. As such at Deepawali Poojan or Luxmi Poojan, Ganesh Poojan is a must.
Though I just reproduced the text as it is from that site, I (please read this as my opinion and my belief) would like to stop believing what ever being read or said. One of the strength of the Hinduism is the vedas which as become its Achilles heel as well due to its limited reach and interpretative nature. Due to this many customs and mores are born and with passage of time these becomes authoritative.
I always believe in Vedas as the authoritative text though I am unable to interpret it. I have to lay my belief on individuals whom I consider as aacharya or gurus. So, I would not believe this explanation given in the site and also do not worship Lakshmi and Ganapathi together as described in the site as well.
For me Hindu worship is completely symbolic and so are all our Gods and Deities. Lakshmi Ganesha being worshipped together means worshipping wealth along with tolerance and patience. Lakshmi signifies Wealth and Ganpati is symbolic for "vivek' and "vighna-harta". If only wealth is accompanied by these characters it will truly fruitify, or else it can lead to devastation. That's my take!