It's not too uncommon for a sitting president to have a "serious" primary challenger from inside his party. Typically, this is more likely when the president has low approval ratings (so an ambitious challenger has a good opportunity) or when a wing of the party thinks the president isn't ideologically aligned with them. I can think of three between the 1970s and 1990s that were significant primary challenges that may have weakened the president and contributed to his failure to be reelected. Plus two bonus ones from earlier that caused the president to withdraw.

The question of whether a "political party" has attempted to primary out a president is a little confused. The primary itself is the method by which the members of the political party make the decision on who should be their nominee. So the political party doesn't "primary out" a president, the candidate (presumably with political support from a portion of party's members) runs in the primary and the party as a whole makes a choice.

Someone runs against the president in almost every reelection campaign. But I don't think your question is about weak or token primary challengers (like Harold Stassen vs. Ronald Reagan in 1984, or Lyndon LaRouche vs. Bill Clinton in 1996), so I'm disregarding them.

Also, I'm limiting this answer to the modern primary system. After about 1968 (which is a whole story unto itself) both parties moved to primary systems that required more binding state primary elections and limited the role of non-elected party insiders. The small-d democratic trend has broadly continued in this direction since then. Before then, primary elections were often non-binding and few states had actual priimary elections, and the nominee was selected more by party insiders (in the proverbial smoke-filled back room).

Anyway, enough caveats and asides:

• 1992: Pat Buchanan vs. George H.W. Bush - Bush was both unpopular (approval in the low 40s, high 30s) and ideologically moderate. Buchanan challenged him as a more conservative candidate, and was significant challenger but didn't win any states. Bush ended up losing to Clinton in the general election.
• 1980: Ted Kennedy vs. Jimmy Carter - Carter had low approval ratings and Kennedy, from a powerful political family and long-expected to run sooner or later, seized the opportunity. International events helped rally support for Carter, and Kennedy didn't run especially well. Carter ended up losing to Reagan in the general.
• 1976: Ronald Reagan vs. Gerald Ford - Reagan was an especially strong primary challenger from a more conservative wing of the Republican party. Ford barely won the primary, then lost to Jimmy Carter in the general election. Reagan went on to run again in 1980 and defeat Jimmy Carter.

Ok, these are before modern primaries, but I can't resist. These two primary challenges led to the sitting president withdrawing from the race:

• 1968: Eugene McCarthy vs. Lyndon Johnson - Johnson faced growing opposition to the Vietnam War. McCarthy did surprisingly well in the first primary, New Hampshire, leading Johnson to withdraw from the race. (For a quick, interesting overview of Johnson's decision to withdraw, check out John Dickerson's Whistlestop podcast called "Retiring for Peace"). Then Robert Kennedy (yep, brother of former President John F. Kennedy and 1980 nominee, Ted Kennedy), and Hubert Humphrey (Johnson's vice president) both jumped in. There were only 13 states running primary elections; but Humphrey joined the race after Johnson quit, and was too late to participate in them. He went after delegates from state conventions. Kennedy was running strong second to Humphrey based on number of delegates, then he got assassinated in June after winning the California primary election. The 1968 Democratic Party convention in Chicago was roiled by protests and riots... a whole big mess. Humphrey was nominated (without running in any primaries) and lost to Nixon in the general election.
• 1952: Estes Kefauver vs. Harry Truman - Truman was facing low approval numbers and growing opposition to the Korean War, and Kefauver beat him in the first primary, New Hampshire. Truman withdrew from the race. There were only 14 states that held primary elections back then, and Kefauver won most of them. However, the party "bosses" (including President Truman) were opposed to Kefauver and ended up nominating Adlai Stevenson... who didn't run in any of the primaries. Stevenson lost to Eisenhower in the general election.

TL;DR

Two immediate questions pop out: 1) Who ran for president and 2) who is “the party” attempting to “primary” the sitting president.

Do you only consider people to have “run for president” if they received primary votes/votes at a convention or do you include everyone who took a serious look at a presidential election. I’ve seen people jump around between these two definitions frequently and both are valid in their own way. For instance, LBJ seriously considered reelection and not running for reelection in 1968 and many factors, including Eugene McCarthy’s “victory” in the New Hampshire Primary (he lost to Johnson but garnered 42% of the vote) forced Johnson out. I think we should clearly count this as LBJ “running” but I’ve often seen this claim implicitly rejected in broader argument about presidential primary challenges. Did Grant “run for president” in 1876? While Grant was not a candidate at the 1876 convention (unlike in 1880) he was a serious contender who felt out running for a third term before bowing out in the face of significant blowback.

http://library.cqpress.com/cqresearcher/document.php?id=cqresrre1927031900

The general interpretation of this letter was that Grant's supporters were at liberty to make it appear the imperative duty of the Republicans to renominate him, and of the President to accept the nomination. The third term boom was demolished soon after the meeting of the first session of the Forty-fourth Congress in December, 1875. William M. Springer, Illinois, a Democratic member, offered a resolution [ratifying the two term precedent saying that departing it would be “unwise, unpatriotic, and fraught with peril to our free institutions”]. The resolution was adopted without debate by the immense majority of 234 to 18, with 38 not voting.

I don’t know if this fits your criteria. The party decided against Grant’s faction’s push for a third term but it didn’t provide an alternative candidate and unlike 1880, Grant bowed out before the actual convention fight.

Who is the party? Your framing of the question assumes the party is finding a challenger to the sitting president but the 19th century provided multiple examples of a very different sort of scenario taking place: a sitting president (who tried and failed to create a third party) be disowned by his own party. John Tyler was literally expelled form the Whig party during his presidential term and Andrew Johnson was both publicly repudiated in the 1868 GOP platform and actually received votes for the Democratic party’s nomination.

There have been multiple cases where incumbent presidents received serious primary challenges (most recently 1948, 1976 and 1980) all featuring a vice presidency who assumed the presidency. 1976 (GOP) and 1980 (Dem) were examples of insurgents tapping into a dissatisfied faction of their party to make a credible shot at dethroning the president. However, in each case, the President won, in large part, because “the party” ultimately stood by them. For instance, according to The Party Decides’ methodology, Carter won 2/3 of “weighted endorsements” versus Kennedy’s 1/3 and Ford was the clear favorite over Reagan in 1976 (e.g. 85% of governors endorsed Ford over Reagan). In both of these cases you could say the party was split but ultimately the party decided on the incumbent. I like this footnote from the party decides

. See Ferrell (1994) for a remarkable example of how a small and informal group of top Democratic party leaders—some elected and some not—engineered Harry Truman’s selection as vice president in 1944.

I haven’t read Ferrell but I think this does a good job showing how Truman fits into this mold.

Perhaps the two clearest examples of the phenomena, John Tyler and Andrew Johnson, were accidental presidents who attempted to create new national parties. In both cases the party didn’t merely attempt to primary the sitting president, they actively blocked them from receiving the nomination. In John Tyler’s case, he was expelled from the Whig party during his one term in office. Tyler was actually nominated as a third party candidate (Democratic-Republicans) on May 27, 1844 running primarily on the issue of Texas’ annexation. He ultimately cut a deal with the Democratic Party, withdrew and endorsed Polk (which Tyler believed won Polk the election).

Millard Fillmore, Chester Arthur (unelected presidents) and Franklin Pierce sought and did not receive renomination. Fillmore had a real shot at nomination unlike Arthur who was not a serious contender. Both Fillmore and Pierce received over 40% of the vote on the initial round before ultimately being defeated (with Fillmore having a better shot than Pierce as renomination). Fillmore and Pierce’s failures also reflected the unsettled status of the current party system during this period. Both were relatively unsuccessful presidents who were unable to reconcile the internal tensions in their party (Skowronek treats Franklin Pierce as the classic “disjunctive” president/presidency).

From this you can see that “unelected” presidents are frequently subject to primary challenges. There are multiple examples of the party rejecting the current president and multiple examples of serious primary challenges (I’ve not included something like Buchanan’s 1992 run which never seriously challenged Bush) but none of these scenarios involved the party rallying around the challenger. The 1840s and 1850s provide multiple examples of the president losing convention fights.