First, it should be made clear that comparing the number of times people are mentioned in the Quran doesn't necessarily show their status. For example, Yusuf (AS) was mentioned by name 27 times in the Quran and the Prophet Muhammad (SAW) was "only" mentioned by name 5 times. This, of course, does not show any superiority of Yusuf (AS) over Muhammad (SAW).
For the question of why Ibrahim (AS) was mentioned in the Quran so many times (169 to be precise), Allah explicitly says in the Quran:
There has already been for you an excellent example in Ibrahim and those with him, when they said to their people, "Indeed, we are disassociated from you and from whatever you worship other than Allah. We have denied you, and there has appeared between us and you animosity and hatred forever until you believe in Allah alone" except for the saying of Abraham to his father, "I will surely ask forgiveness for you, but I have not [power to do] for you anything against Allah. Our Lord, upon You we have relied, and to You we have returned, and to You is the destination. (60:4)
In this verse, Allah mentions a specific instance of a good example, but the point is that Ibrahim (AS) was a great example for everyone after him. He was an extremely devout person who submitted to God and went against his own people at a very young age for Allah's cause.
In old age, he continued to show his complete submission to God by passing the many tests Allah gave him including one where he was asked to sacrifice and slaughter his own son. This is why Allah promised Ibrahim (AS):
"Indeed, I will make you a leader (Imam) for the people." (2:214)
As part of that promise, Ibrahim (AS)'s stories, demonstrations of faith, and blessings are mentioned to us so that we follow his example since he is our Imam. Ibrahim (AS)'s example was so good that Allah said:
And who would be averse to the religion of Abraham except one who makes a fool of himself. And We had chosen him in this world, and indeed he, in the Hereafter, will be among the righteous. (2:130)
Here are some things in our religion which come from Ibrahim (AS) directly or indirectly:
The Holiness of Makkah: It is because of the dua of Ibrahim (AS) to make it so. See 2:126
The Kaabah: He built the Kaabah which is our Qibla. See 2:127
The Hajj and its rituals: See 22:27 and 2:128
Eid ul-Adha: Comes from the sacrifice of Ismail (AS) commanded to Ibrahim (AS) which he fulfilled in his complete submission to God. Then, Allah replaced Ismail (AS) with an animal to sacrifice instead. See 37:102
The Prophet Muhammad (SAW): He was the answer to a dua of Ibrahim (AS). See 2:129
The countless other prophets from his progeny in Bani Israel: They were part of Allah's reward to him for his submission and patience. See 29:27