"Barbara Bush was the first lady of the greatest generation", Meacham said.
The former president is known for wearing colorful socks.
The woman died at the age of 92 years.
President Trump excused himself from attending, tweeting Saturday: 'Heading to the Southern White House to watch the Funeral Service of Barbara Bush. He was among four former USA presidents at the service.
Mrs. Bush will be buried in College Station with her daughter.
Barbara Bush was buried Saturday afternoon at her husband's presidential library at Texas A&M University in College Station, about 100 miles (161 kilometers) northwest of Houston.
Other guests included former Rep. She died of leukemia in 1953. Hundreds of people lined both sides of the street near the campus ahead of the service. She taught them to "say please and thank you", he said, and to "be kind, always tell the truth, never disparage anyone, serve others, treat everyone as you would want to be treated". Clinton did speak at a graveside service for her at Arlington National Cemetery in 1994.
After seeing how many people had lined up to pay their respects to his wife, former President George H.W. Bush made a decision to attend - he sat at the front of the church in a wheelchair, offering his hand and smiled as people shook it, for about 15 minutes.
They stopped along the way to shake hands, as mourners sang "Joyful, joyful, we adore thee", which Barbara Bush had requested as the final song. Bush Sr. followed closely behind in a wheelchair that was pushed by his son.
Jeb Bush said his father appeared a bit disheveled but sat by his mother's bed and held her hand. Numerous women wore the former first lady's favorite color, blue, and trademark pearls.
"I'm sick of him", she said, with a quick flash of anger in her eyes.
The ceremony was "a very brief but poignant and attractive ending to a very moving and incredible day", family spokesman Jim McGrath told The Associated Press.
The White House said in a statement that Trump will not attend the service "to avoid disruptions due to added security, and out of respect for the Bush Family and friends attending the service".
Barbara Bush was remembered as the "first lady of the Greatest Generation" during a funeral Saturday attended by four former US presidents and hundreds of other people who filled a Houston church with laughter as much as tears, with many recalling her quick wit and devotion to family.
Baker spoke about their friendship and how Bush helped her when she moved to the area.
Baker emphasized Barbara Bush's kindness, compassion and ability to make others feel welcomed. "He may not be able to keep a job, but he's certainly not boring.'", said friend Susan Baker.
Hundreds traveled from across the United States to attend the funeral of a woman who was as distinguished as she was down to earth.
Presidential historian Jon Meacham eulogized her as "The First Lady of the Greatest Generation".
Meacham recalled her work bringing awareness to AIDS patients and in promoting literacy.
Numerous speakers made sure to include amusing stories about Bush, who was known for being quick-witted and having a good sense of humor.
Following the service, the casket was carried out of the sanctuary by Bush's eight grandsons. "Today, my thoughts and prayers are with the entire Bush family", Trump tweeted.
At St. Martin's Church on Saturday, the focus was on a different kind of presidential.
A river of bad blood flows between Trump and the Bushes, from Trump's repeated bashing of Jeb Bush as "low energy" to George H.W. Bush's dismissal of Trump as a "blowhard".
Along with the former president, three other former presidents also will attend: the couple's son, George W. Bush, Bill Clinton and Barack Obama.
Jeb Bush, who delivered one of the eulogies, said it was a blessing to have a teacher like his mother.
First Lady Melania Trump.
Trump's absence from Bush's funeral is not uncommon for a sitting president.
First lady Melania Trump personally invited George Hannie, former White House maÎtre d' in the Bush White House, and Buddy Carter, the current White House usher, who also worked for the Bushes, to join her at the service, according to her press secretary, Stephanie Grisham.
Bush, who died Tuesday at 92, embodied the political establishment.
Some 1,500 guests are expected to attend a private funeral for Barbara Bush at the nation's largest Episcopal church. The priest is a longtime friend of the Bush family.
Thousands of people on Friday paid respects to Barbara Bush, wife of the nation's 41st president and mother of the nation's 43rd.